Historical context and the two obstructions

Hooley proved Artin’s predicted asymptotic under the Riemann hypothesis for the Dedekind zeta functions of the Kummer fields \(\mathbb{Q}(\mu_n,a^{1/n})\), with \(n\) squarefree [13]. These fields encode the obstructions to maximal order. For a prime \(q\) and \(p\nmid aq\), the conditions \[p\equiv1\pmod q,\qquad a^{(p-1)/q}\equiv1\pmod p\] say that \(p\) splits completely in \(\mathbb{Q}(\mu_q,a^{1/q})\). They hold precisely when \(q\) divides the index \((p-1)/\mathop{\mathrm{ord}}_p(a)\). Hooley’s method counts and sums such splitting conditions. Here a restricted factorization of \(p-1\) confines the possible index divisors to ranges where they can all be removed.

Classical unconditional results allow a finite choice of bases. Gupta and Murty constructed a set of thirteen integers from three distinct primes, at least one of which is a primitive root modulo infinitely many primes [9]. Heath-Brown proved that at most two prime bases fail the infinitude assertion, so at least one of \(2,3,5\) satisfies it [11]. Averaging over the base gives another approach. Goldfeld and Stephens established early average theorems [7, 26]; Klurman, Shparlinski, and Teräväinen obtained the expected asymptotic for almost all integers in a range as short as \(1\le a\le\exp((\log\log x)^2)\) [17]. The exceptional set may depend on \(x\), so these average results do not by themselves settle a prescribed base.

Two difficulties must be addressed for a fixed base. One is analytic: the splitting fields vary with the possible index divisor. A standard simultaneous zero-free region for Hecke characters associated with an abelian extension has logarithmic width depending on the field, conductors, and height, and permits at most one simple real exceptional zero [28]. The common width in Theorem 2 instead gives a common power saving. The other difficulty is the prime construction: we need many primes for which the possible index divisors fall in the range controlled by that saving. We now describe the two parts of the proof and the points where they meet.

The Hecke zero-free argument

Fix a cyclotomic field \(F\) containing \(\mu_{12}\) and a finite-order Hecke character \(\eta\). The analytic construction uses cubic theta coefficients under sextic Kummer twists. Patterson’s cubic theta series over \(\mathbb{Q}(\sqrt{-3})\) is the model for the Gauss coefficients and their transformations [24, 25]; the comparison formulas are recorded by Dunn and Radziwiłł [5]. The companion manuscript The Quasi-Riemann Hypothesis develops a coefficient reflection and an additive probe over that field while retaining unrestricted cube factors [23]. Its zero-free theorem treats Dirichlet \(L\)-functions and finite-order Hecke \(L\)-functions over \(\mathbb{Q}(\sqrt{-3})\). We construct the required mechanisms over each fixed \(F\), using the cubic theta representation of Kazhdan and Patterson and a normalized global Whittaker factorization consistent with their correction [15, 16].

Two local calculations govern the comparison. We choose generators for which sextic reciprocity has no supplementary factor. In a reflected interaction between a prime occurring to valuation one in the varying twist and a varying squarefree coefficient prime, the sextic exponents are \(1\) and \(-4\). Their sum is \(3\) modulo \(6\), so the interaction is quadratic. The quadratic large sieve of Heath-Brown, in its number-field form due to Goldmakher and Louvel, then controls the reflected moment [12, 8].

On the Poisson side, a relative-norm formula identifies the cube of the cubic Gauss phase with the negative of a unitary Hecke-character value. A finite-order correction gives the target character \(\eta\). At a prime ideal \(\mathfrak p\) outside a fixed excluded set, the first squarefree theta term at the principal frequency is \(-\eta(\mathfrak p)(\mathrm{N}\mathfrak p)^{-\xi}\), where \(\xi\) is the Mellin variable for the coefficient norm. Together with the constant term, it supplies the local factor of the reciprocal target \(L\)-function. The remaining local terms give a holomorphic, nonvanishing correction in the region used at the end of the proof.

Reflection and Poisson summation thus estimate the same additive average in two ways. A conductor large sieve and zero detection control the nonprincipal frequencies. The resulting power saving for the principal inverse Mellin integral continues its Mellin transform across the line \(\operatorname{Re}s=1\) and excludes zeros of the target function. The use of ideal norms keeps the numerical exponents in this argument independent of the degree of \(F\). Sections 3–8 carry out this proof; Appendix 13 provides the theta normalization.

Controlled predecessors in a fixed progression

The construction produces many primes of the form \[p=crQ+1,\qquad c\in\{2,4\},\qquad Q>x^{0.9}\text{ prime},\] where every prime divisor of \(r\) lies between \(\exp((\log x)^{0.1})\) and \(\exp((\log x)^{0.3})\). We also prescribe a residue class on which the Legendre symbol \((a/p)\) equals \(-1\). This excludes \(2\) as an index divisor. A nontrivial remaining index has a prime divisor in \(rQ\): the factor \(Q\) leads to a small elementary exceptional set, and the factors of \(r\) fall within the uniform splitting estimate.

The construction starts with weighted integers \(d=crQ+1\) of this shape. A sieve removes small prime factors of \(d\). For a remaining composite \(d=mn\), with \(n\) its least prime factor, a marked bilinear estimate replaces the restriction on the cofactor \(m\) by an explicit density and a weaker condition on its small prime factors. Buchstab’s least-prime-factor identity leaves a positive margin between the initial sifted mass and the mass removed in this comparison. A separate upper bound makes the contribution from two prime factors close to \(\sqrt x\) smaller than that margin.

The marked prime estimate is an input from The Poisson–Dirichlet law for the prime factors of \(p-1\) [21]. We extract its scalar coefficient criterion and verify the full character and frequency estimates for the rough-factor comparison. The weights require \(d-1\) to have a prime divisor in each of several prescribed ranges; these choices are the marks. A bounded multiplier in the bilinear estimate may depend on the quotient left after every power of these primes has been removed. A congruence condition on \(d=mn\) need not have this invariance. We therefore expand \(mn\equiv u\pmod M\) in Dirichlet characters and separate it into factors \(\lambda(m)\lambda(n)\) in the two scalar coefficient sequences. Fixed twists preserve the required long character-sum bound.

The resulting fixed-progression construction is stated separately in Proposition 3. It applies to any fixed data \(M,c,u\) satisfying its local coprimality conditions, independently of the primitive-root application. Its proof uses Bombieri–Vinogradov distribution, a Brun–Hooley block sieve, and the continuous rough-number density [3, 29, 6, 4]. The precise elementary sieve and density forms are adapted from [20], with proofs included here.

Organization and notation

Section 2 states the controlled-predecessor and splitting estimates and combines them to prove Theorem 1. Sections 3–8 prove Theorem 2 for a fixed arbitrary \(F\) and \(\eta\), independently of the integer base. The analytic scale parameters in those sections are local to that proof. Section 9 returns to the fixed integer \(a\) and derives the uniform splitting bound. Section 10 proves the marked coefficient and rough-factor estimates; Section 11 supplies the elementary sieve facts; Section 12 proves the fixed-progression construction. Appendix 13 verifies the selected Kazhdan–Patterson specialization and Whittaker normalization.

Throughout, \(\log\) is the natural logarithm, and \(L=\log x\) in the prime-counting arguments. The symbols \(\ll\) and \(O\) have their usual meaning; subscripts record permitted dependence when it matters. Auxiliary parameters are fixed before the relevant asymptotic variable tends to infinity. Uniformity in those parameters is stated explicitly. The general representation-theoretic, class-field, Hecke-continuation, and classical sieve theorems are cited at their points of use. The marked Type II argument also uses the companion papers cited in the bibliography.

The automorphic input and its exterior coefficients

We use the determinant-modified cover of Kazhdan and Patterson with parameter \(c=2\), with the ordinary torus section and the canonical root splittings. The embedding \(\iota:\mu_3(F)\hookrightarrow\mathbb{C}^\times\) is oriented as in the exterior cocycle formula below. If \(a_v(y)\) denotes the chosen torus lift, define the effective scalar \(\mathcal C_v(x,y)\) by \[\pi(a_v(x))\pi(a_v(y)) =\mathcal C_v(x,y)\pi(a_v(xy))\] in a genuine representation \(\pi\). Thus this notation includes the chosen embedding of the central \(\mu_3\).

In (28), the condition \(W_{\mathfrak p}(1)=1\) fixes every exterior spherical scalar; the remaining normalization belongs to the \(S\)-functional. 13 derives this product from the nonzero global functional and local uniqueness, including the correction to the original global formula. The noncompact norm integrations below require the separate absolute bounds in 16.

Thus theta supplies the exterior factor in \(B_m\). The ideal character will enter through an \(S\)-average, and finite upper-unipotent translations will insert the moving sextic twist. The chosen \(S\)-Whittaker function first supplies a radial profile; an exact radial average will replace it by \(P_G\). We prepare those \(S\)-components next.

The arithmetic power classes also describe the torus factor at \(S\). Put \(\mathcal C_S=\prod_{v\in S}\mathcal C_v\). At an exterior prime, [eq:theta-effective-cocycle] and the tame Hilbert-symbol formula give \(\mathcal C_{\mathfrak p}(x,y)=(x,y)_{6,\mathfrak p}^{2}\) for all local \(x,y\). For rational \(a,b\), the product formulas for the chosen torus lifts and for the Hilbert symbol therefore imply \[\mathcal C_S(a,b) =\prod_{\mathfrak p\notin S}\mathcal C_{\mathfrak p}(a,b)^{-1} =H_S(a,b)^2.\] Both sides are continuous on \(F_S^\times\times F_S^\times\), so weak approximation gives \[ \mathcal C_S(x,y)=H_S(x,y)^2 \qquad(x,y\in F_S^\times). \tag{35}\] In particular this factor depends only on the classes in the fixed finite group \(\mathcal V=F_S^\times/F_S^{\times6}\). The local sections here are the same ones used by the rational splitting.

Bounds and a prescribed finite Whittaker function

For \(v\mid\infty\), define the derivative in the logarithm of the norm by \[\mathcal D_v f(y) =\left.\frac{\,\mathrm{d}}{\,\mathrm{d}t}f(e^{t/2}y)\right|_{t=0}.\] The factor \(1/2\) makes \(|e^{t/2}y|_v=e^t|y|_v\).

We give both the Bessel calculation and bounds for the fixed rotations of the vectors that we will use. Choose complex coordinates in which \(\psi_v(z)=e^{i c_v\operatorname{Re}z}\), with \(c_v>0\); a harmless rotation and a change of sign give this form. Put \(\varrho=|y|_v^{1/2}\). In the dominant normalized principal series set \(\nu_0=1/2+1/6=2/3\). Its spherical section, restricted to \(wn(x)a(y)\), is a constant times \[\varrho^{2\nu_0}(\varrho^2+|x|_{\mathrm{ord}}^2)^{-2\nu_0},\] where \(|x|_{\mathrm{ord}}\) is the ordinary complex modulus. The Jacquet integral is absolutely convergent because \(2\nu_0>1\). Using the Laplace representation of the negative power and the two-dimensional Gaussian Fourier integral gives \[\begin{align*} \varrho^{2\nu_0}\int_{\mathbb{R}^2} \frac{e^{-ic_v x_1}}{(\varrho^2+x_1^2+x_2^2)^{2\nu_0}} \,\,\mathrm{d}x_1\,\mathrm{d}x_2 &=\frac{\pi \varrho^{2\nu_0}}{\Gamma(2\nu_0)} \int_0^\infty t^{2\nu_0-2} e^{-t\varrho^2-c_v^2/(4t)}\,\,\mathrm{d}t\\ &=c'_v \varrho K_{2\nu_0-1}(c_v\varrho) =c'_v \varrho K_{1/3}(c_v\varrho),\qquad c'_v\ne0. \end{align*}\] The Laplace and Gaussian integrals may be interchanged absolutely: after integrating the Gaussian without its phase the integrand at zero is \(O(t^{2\nu_0-2})\), with exponent greater than \(-1\). This proves (33).

For an upper root translate, Whittaker equivariance gives \[ W_{\pi(n(z))\varphi}(a(y))=\psi_v(yz)W_\varphi(a(y)). \tag{36}\] Suitable complex linear combinations of the two real Lie derivatives in \(z\) therefore multiply the spherical function by \(y\) and by \(\bar y\), respectively. Write \(\omega_m(y)=(y/|y|_v^{1/2})^m\) for \(m\in\{1,-1\}\). After multiplying each derivative vector by a nonzero constant, its Whittaker function is \[ W_{v,m}(a(y)) =|y|_v\,\omega_m(y)K_{1/3}(c_v|y|_v^{1/2}). \tag{37}\]

Here are uniform bounds for every vector in a fixed finite list consisting of these derivative vectors and their fixed compact rotations. A rotation of a first Lie derivative of the spherical vector is another first Lie derivative, since the spherical vector is compact invariant and \(\pi(k)\,\mathrm{d}\pi(X)=\,\mathrm{d}\pi(\operatorname{Ad}(k)X)\pi(k)\). For a fixed real Lie algebra element \(X\), differentiate the spherical section using \[f^\circ(g)= \left(\frac{|\det g|_{\mathrm{ord}}} {|g_{21}|_{\mathrm{ord}}^2+|g_{22}|_{\mathrm{ord}}^2}\right)^{2\nu_0}.\] On \(wn(x)a(y)\), its first Lie derivative and any number of \(\mathcal D_v\)-derivatives are the preceding spherical integrand times a finite sum of bounded homogeneous rational functions of \((y,x)\). More precisely, for every norm derivative order \(b\) and every \(x_1\)-derivative order \(h\), the resulting derivative is bounded by \[C_{b,h} \varrho^{2\nu_0} (\varrho^2+|x|_{\mathrm{ord}}^2)^{-2\nu_0-h/2}.\] This follows by differentiating the displayed quotient: each derivative adds a polynomial numerator of the same degree as its denominator, and an \(x_1\)-derivative lowers the homogeneous degree by one. The integral of the bound with \(h=0\) is \(O(\varrho^{2-2\nu_0})=O(\varrho^{2/3})\). For \(\varrho\ge1\), integrating by parts \(h\) times against \(e^{-ic_vx_1}\) instead gives \(O_{b,h}(\varrho^{2/3-h})\). These integrable derivative bounds justify the integrations by parts by cutoff and limit. Consequently, for every integer \(j\ge0\) and every \(A>0\), the fixed list satisfies \[ |\mathcal D_v^j W(a(y))| \le C_{A,j}\min\bigl(|y|_v^{1/3},|y|_v^{-A}\bigr). \tag{38}\]

At a finite place the corresponding elementary bounds hold for every fixed smooth vector. To be explicit, let \(W_\varphi(y)=\lambda_v(\pi_v(a(y))\varphi)\). There is an additive open subgroup \(J\subset F_v\) for which \(n(J)\) fixes \(\varphi\). The identity \[W_\varphi(y)=\psi_v(yx)W_\varphi(y)\qquad(x\in J)\] forces \(W_\varphi(y)=0\) when \(|y|_v\) is sufficiently large. For the other end, put \(Q_v=\#(\mathcal O_v/\mathfrak p_v)\) and \(z_v=a(\varpi_v^3)\). This element is central in the covering torus. By the two-cell Jacquet calculation [15], the two constituents of the unnormalized ordinary Jacquet module of the principal series have \(z_v\)-eigenvalues of absolute values \[Q_v^{-1}\quad\hbox{and}\quad Q_v^{-2}.\] These are the exponents \(1/2-1/6\) and \(1/2+1/6\) on a triple valuation step. Exactness of the ordinary Jacquet functor gives the same possible eigenvalues on the exceptional subquotient. Hence a polynomial all of whose roots have those absolute values annihilates the action of \(z_v\) on this Jacquet module.

For completeness, this gives a recurrence on the Whittaker values themselves. If \(P\) is such an annihilating polynomial, then \[P(\pi_v(z_v))\varphi =\sum_{i=1}^h\bigl(\pi_v(n(x_i))-1\bigr)\varphi_i\] for finitely many \(x_i,\varphi_i\). Apply \(\lambda_v\pi_v(a(y))\). For all sufficiently small \(|y|_v\) the right side is zero because every \(\psi_v(yx_i)\) is one. Cube cocycles are trivial, so the left side is the recurrence \(P(E)W_\varphi(y)\), where \(E W_\varphi(y)=W_\varphi(y\varpi_v^3)\). On each valuation progression its solutions are sums of a polynomial in the valuation times the powers of the roots of \(P\). The unit variable ranges over only finitely many classes after restricting to a sufficiently small unit subgroup which fixes the vector and consists of cubes. It follows that, for every fixed \(\delta<1/3\), \[ W_\varphi(y)=0\quad(|y|_v>C_\varphi),\qquad |W_\varphi(y)|\le C_{\varphi,\delta}|y|_v^\delta \quad(|y|_v\le1). \tag{39}\] All constants in this paragraph may depend on the place, the vector, and the functional. For a fixed finite list one may choose a common positive \(\delta\). The local twisted Jacquet quotients of \(\Theta_v\) are finite dimensional by the quotient argument in 13.2. Thus the same conclusions hold term by term for the finite sum of products at \(S\) in 13.

Here is the choice at \(S\) that will be used later. Write \(S_f=S\setminus S_\infty\). Let \(\mathscr L\) be any fixed finite family of smooth unitary characters \(\mathcal L_S\) of \(F_S^\times\), each trivial on \(U_S\), radially trivial at infinity, and with angular component \(\omega_{m_v}\), \(m_v\in\{1,-1\}\), at each complex place. This permits the arithmetic character \(L_S\) and its finitely many finite-order twists used below. Factor the nonzero global functional at its complex places, leaving a nonzero functional at the finite places of \(S\). Apply 15 with an open subgroup contained in the sixth powers and in the intersection of the kernels of the finite components of all \(\mathcal L_S\in\mathscr L\). We obtain one product \(U=\prod_{v\in S_f}U_v\) and one finite vector. For each \(\mathcal L_S\), choose at each complex place the derivative of its angular type. The functions in (37) are nonzero at one, and their normalization gives an \(S\)-Whittaker function \[ W_{S,\mathcal L_S}(a(y))=\mathbf{1}_U(y_f) \prod_{v\mid\infty}|y_v|_v\,\omega_{m_v}(y_v) K_{1/3}(c_v|y_v|_v^{1/2}). \tag{40}\] In particular \(\mathcal L_S(y)^{-1}W_{S,\mathcal L_S}(a(y))\) is the product of the same radial Bessel factors and \(\mathbf{1}_U(y_f)\) for every member of the family. The radial profile and its Gaussian averaging kernel below may therefore be chosen in common for this finite family.

The complex constant term is radial by 13, whereas \(\mathcal L_S^{-1}\) has a nontrivial angular component. We will use this distinction in 6, after the actual weighted theta sums have been shown to descend to the quotient on which they are integrated.

Integration on norm sheets

Recall that \(r=\#S_\infty=[F:\mathbb{Q}]/2\), and define \[X_u=\{y\in F_S^\times:|y|_S=u\}\qquad(u>0).\] For \(d\in F_S^\times\), put \(q_d=|d|_S\). Multiplication by \(d\) sends \(X_l\) to \(X_{q_dl}\), and inversion sends \(X_l\) to \(X_{1/l}\). The forward and reflected theta terms will therefore lead to sheet integrals at norms \(q_dl\) and \(q_d/l\), respectively. We now bound such one-variable profiles, including absolute envelopes that will justify the ideal sums and the radial average. The two signed orbit substitutions will be proved with their actual integrands in 6.

There is at least one complex place. Use the norm section \(\ell_u\) from 3; it satisfies \(|\ell_u|_S=u\) and \(\ell_u\ell_v=\ell_{uv}\). Choose multiplicative Haar measures and Haar measure on \(X_1\) so that \[ \int_{F_S^\times}f(y)\,\,\mathrm{d}^\times y =\int_0^\infty\int_{X_u}f(y)\,\,\mathrm{d}^\times_u y\,\frac{\,\mathrm{d}u}{u}, \tag{41}\] where the measure on \(X_u\) is the translate by \(\ell_u\) of that on \(X_1\). At a complex place we take angle measure of total mass one, so its multiplicative measure is a fixed constant times \(\,\mathrm{d}t/t\) times angle measure, with \(t=|y_v|_v\). At a finite place every valuation shell has the same multiplicative measure.

The lemma also gives the smooth separation on norm dyads that we will use. If \(\rho\) is a fixed smooth compactly supported function on \(\mathbb{R}\), integration by parts twice in its ordinary Fourier transform gives \[ \left\|\widehat{\rho(t)\Phi_\kappa(u e^t)}\right\|_{L^1(\mathbb{R})} \le C_{A,\rho}\min(u^\alpha,u^{-A}). \tag{45}\] Indeed the \(L^1\) norm is bounded by a constant times the suprema of the function and its derivatives of orders one and two; these derivatives are controlled by (43). Fourier inversion with \(t=\log x-\log y\), for \(x,y\) in fixed ratio intervals, therefore separates the two norm variables with bounded \(L^1\) cost and retains any chosen large-\(u\) power saving.

Exact radial smoothing

For the chosen \(S\)-function (40), the angular characters cancel against \(\mathcal L_S^{-1}\). Define its radial profile by \[ V(u)=\int_{X_u}\mathbf{1}_U(y_f) \prod_{v\mid\infty}|y_v|_v K_{1/3}(c_v|y_v|_v^{1/2})\,\,\mathrm{d}^\times_u y. \tag{46}\] It is a positive, nonzero profile covered by 16. In this subsection the Mellin convention is \(\widehat f(s)=\int_0^\infty f(u)u^s\,\,\mathrm{d}u/u\). The unreflected theta average will contain \(V(q_{\mathfrak A}l)\). At \(l=T/Z\), the next lemma averages this profile to exactly \(P_G(q_{\mathfrak A}/Z)\). After the Weyl operation the norm variable is inverted, and the same kernel instead produces the controlled profiles \(\int\mathcal K(T)H_\kappa(u/T)\,\,\mathrm{d}T/T\).

An automorphic expression for \(B_m\)

All implied constants in this section may depend on the fixed field, \(S\), the target character \(\eta\), the compact set \(\mathscr K\), and the chosen finite family of local theta data.

We first replace elements by ideal rows. Choose a balanced allowed generator \(m_0\) of each exterior ideal \(\mathfrak m\). For every nonzero \(m\in R\) generating \(\mathfrak m\), write \(m=\varepsilon_m m_0\) with \(\varepsilon_m\in U_S\). When \(m\in\mathscr B_Z(\mathscr K)\), the balanced shapes of \(m_0\), the fixed shape set \(\mathscr K\), and \(q_{\mathfrak m}\asymp_{\mathscr K}Z^{M_0}\) put \(\varepsilon_m\) in a fixed compact subset of the norm-one group. The \(S\)-unit theorem makes \(U_S\) discrete there, so one ideal is generated by only boundedly many elements of the row set. Put \[\Gamma=U_S^6,\qquad U_6=U_S/\Gamma,\qquad \upsilon_m=\varepsilon_m\Gamma\in U_6.\] The group \(U_6\) is finite. The value of \(B_m\) depends on the unit \(\varepsilon_m\) only through \(\upsilon_m\), which is fixed when the row \(m\) is fixed.

For a nonzero row let \(j_{\mathfrak p}=v_{\mathfrak p}(\mathfrak m)\bmod6\) in \(\{0,\ldots,5\}\). For an allowed generator \(A\) of an integral exterior ideal \(\mathfrak A\), reciprocity gives \[ \chi_{\mathfrak A}(m) =\chi_{\mathfrak A}(\varepsilon_m) \prod_{\mathfrak p\mid\mathfrak m} \chi_{\mathfrak p}(A)^{j_{\mathfrak p}}. \tag{53}\] For coprime ideals this is the multiplicative reciprocity law from the arithmetic section. At a shared prime both sides vanish by zero extension, including when \(j_{\mathfrak p}=0\); the zeroth power there is the unit indicator.

Define \(\lambda_{\varepsilon_m}(y)=H_S(\varepsilon_m,y)^{-1}\) on \(F_S^\times\). By [eq:unit-symbol], its value at an allowed generator \(A\) is \(\chi_{\mathfrak A}(\varepsilon_m)\), and isotropy of \(E\) makes it trivial on \(U_S\). It depends only on \(\upsilon_m\). With the character \(L_S\) from [eq:vartheta-s], put \(L_{\upsilon_m}=L_S\lambda_{\varepsilon_m}\). Then \[ L_{\upsilon_m}(A) =\vartheta(\mathfrak A)\chi_{\mathfrak A}(\varepsilon_m) \qquad\text{for every allowed generator }A. \tag{54}\] These characters form a fixed finite family. They are smooth, unitary, and trivial on \(U_S\); their radial components at infinity are trivial, and their angular component at each complex place has power \(1\) or \(-1\).

We insert the moving twists by finite upper-unipotent translations. At an exterior prime \(\mathfrak p\), write \(k_{\mathfrak p}=\mathcal O_F/\mathfrak p\) and use its allowed generator \(p\) as a local uniformizer. For \(1\le j\le5\) put \[\tau_{\mathfrak p,j} =\sum_{v\in k_{\mathfrak p}^{\times}} \chi_{\mathfrak p}(v)^{-j}\psi_{\mathfrak p}(v/p).\] The character is nontrivial, so \(\lvert\tau_{\mathfrak p,j}\rvert=q_{\mathfrak p}^{1/2}\). Define \[ \alpha_{\mathfrak p,j}(v)= \begin{cases} \tau_{\mathfrak p,j}^{-1}\chi_{\mathfrak p}(v)^{-j}, &1\le j\le5,\ v\ne0,\\ 0, &1\le j\le5,\ v=0,\\ 1-q_{\mathfrak p}^{-1},&j=0,\ v=0,\\ -q_{\mathfrak p}^{-1},&j=0,\ v\ne0. \end{cases} \tag{55}\] For every \(t\in\mathcal O_{\mathfrak p}\), \[ \sum_{v\in k_{\mathfrak p}}\alpha_{\mathfrak p,j}(v) \psi_{\mathfrak p}(tv/p) =\chi_{\mathfrak p}(t)^j. \tag{56}\] For \(j\ne0\), substitution by \(t^{-1}\) proves this for a unit \(t\), and both sides vanish for a nonunit. For \(j=0\), the left side is \(1-q_{\mathfrak p}^{-1}\sum_v\psi_{\mathfrak p}(tv/p)\), the unit indicator. Thus the formula includes the masked zeroth power.

For each \(\upsilon\in U_6\), choose the theta vector in 13 whose finite \(S\)-Whittaker function is the indicator of the common small product subgroup contained in sixth powers and in \(\ker L_\upsilon\), and whose complex angular type is that of \(L_\upsilon\). Denote its automorphic function by \(\theta_\upsilon\). The cutoff and the normalizations are those of [eq:theta-chosen-S-function]; these choices form a fixed finite family. For the remainder of this subsection fix a nonzero \(m\), put \(\upsilon=\upsilon_m\), and write \(L=L_\upsilon\).

For \(y\in F_S^\times\) and digits \(\boldsymbol v=(v_{\mathfrak p})_{\mathfrak p\mid\mathfrak m}\), let \(g(y,\boldsymbol v)\) have component \(a(y)\) at \(S\), component \(n(v_{\mathfrak p}/p)\) at \(\mathfrak p\mid\mathfrak m\), and identity at the other exterior places. Use the lifts in [eq:theta-lift-identities], choose representatives of the residue digits, and set \[\boldsymbol\alpha_m(\boldsymbol v) =\prod_{\mathfrak p\mid\mathfrak m} \alpha_{\mathfrak p,j_{\mathfrak p}}(v_{\mathfrak p}), \qquad w_+=1,\quad w_-=w.\] The combined digit sums that we shall integrate are \[ \begin{split} \mathscr D_m^\pm(y) &=\sum_{\boldsymbol v}\boldsymbol\alpha_m(\boldsymbol v) \theta_\upsilon(w_\pm g(y,\boldsymbol v)),\\ \mathscr D_{m,N}^\pm(y) &=\sum_{\boldsymbol v}\boldsymbol\alpha_m(\boldsymbol v) \theta_{\upsilon,N}(w_\pm g(y,\boldsymbol v)). \end{split} \tag{57}\] Rational automorphy gives \(\mathscr D_m^+=\mathscr D_m^-\). We next show that all four sums descend under \(\Gamma\).

For \(\gamma\in\Gamma\), choose representatives \(v'_{\mathfrak p}\equiv\gamma v_{\mathfrak p}\pmod{\mathfrak p}\), and put \[u_{\mathfrak p} =(\gamma v_{\mathfrak p}-v'_{\mathfrak p})/p\in\mathcal O_{\mathfrak p}.\] Let \(k_S=1\), \(k_{\mathfrak p}=n(u_{\mathfrak p})a(\gamma)\) for \(\mathfrak p\mid\mathfrak m\), and \(k_{\mathfrak q}=a(\gamma)\) at the other exterior places. Because \(\gamma\) is an exterior unit, these are compact lifts fixing the original exterior spherical vector. At \(S\) the torus factors involving \(\gamma\) are trivial, since it is a sixth power. At a row prime the exact root relation gives \[n(v'_{\mathfrak p}/p)n(u_{\mathfrak p})a(\gamma) =a(\gamma)n(v_{\mathfrak p}/p).\] The rational diagonal and Weyl lifts therefore satisfy \[ \begin{split} g(\gamma y,\boldsymbol v') &=a(\gamma)g(y,\boldsymbol v)k^{-1},\\ w g(\gamma y,\boldsymbol v') &=a(\gamma^{-1})w g(y,\boldsymbol v)k^{-1}. \end{split} \tag{58}\] The second identity has the same compact factor at the far right.

The upper constant term has the required invariance at these right arguments as well. For \(q\in F^\times\), any adelic \(h\), and any such exterior compact \(k\), the root relation and the product formula give \[ \begin{split} \theta_{\upsilon,N}(a(q)hk^{-1}) &=\int_{F\backslash\mathbb{A}_F} \theta_\upsilon(a(q)n(x/q)hk^{-1})\,\,\mathrm{d}x\\ &=\theta_{\upsilon,N}(h). \end{split} \tag{59}\] Indeed \(n(x)a(q)=a(q)n(x/q)\); rational automorphy removes \(a(q)\), and right invariance removes \(k^{-1}\) at the far right. The substitution \(x\mapsto x/q\) preserves Haar measure on \(F\backslash\mathbb{A}_F\). This argument applies in particular when \(h_{\mathfrak p}=wn(v/p)\). Finally \(\chi_{\mathfrak p}(\gamma)=1\), so \(\alpha_{\mathfrak p,j}(\gamma v)=\alpha_{\mathfrak p,j}(v)\), including the zero digit. Reindexing by \(\boldsymbol v'=\gamma\boldsymbol v\) in [eq:reflection-quotient-lifts,eq:reflection-constant-term-descent] proves the asserted descent of the combined sums before and after \(w\).

Recall \(X_l=\{y\in F_S^\times:|y|_S=l\}\) and the sheet measures from [eq:sheet-disintegration]. The quotient \(X_l/\Gamma\) is compact; choose a measurable fundamental set \(\Omega_1\subset X_1\) and put \(\Omega_l=\ell_l\Omega_1\), using the norm section from that disintegration. Every integral over \(X_l/\Gamma\) means the integral over \(\Omega_l\) with the restricted sheet measure, equivalently the quotient measure induced using counting measure on \(\Gamma\). The factors \(\mathbf{1}_B([y])\) and \(L(y)^{-1}\) also descend, since \([\gamma]=1\) in \(\mathcal V\) and \(L(\gamma)=1\). The resulting integrands are integrable on the quotient: the digit sums are finite and the factors are smooth or locally constant.

Both combined constant terms have zero weighted integral. To see this, rotate \(y\) by a complex number \(z\) of modulus one at one complex place. In the plus sum this acts by the left diagonal \(a(z)\); in the minus sum it acts by \(a(z^{-1})\). The constant term belongs to the torus induction in 13. Its complex inducing law is radial for every right argument, so both sums are unchanged, including at the exterior arguments \(wn(v/p)\). The class condition is unchanged because \(\mathbb{C}^\times\) is sixth-divisible, while \(L(y)^{-1}\) transforms by a nontrivial angular character. Haar invariance therefore gives \[ \int_{X_l/\Gamma}\mathbf{1}_B([y])L(y)^{-1} \mathscr D_{m,N}^\pm(y)\,\,\mathrm{d}_l^\times y=0. \tag{60}\] We may now define the quotient average \[ \begin{split} \mathcal T_m(l) &=\int_{X_l/\Gamma}\mathbf{1}_B([y])L(y)^{-1} \mathscr D_m^+(y)\,\,\mathrm{d}_l^\times y\\ &=\int_{X_l/\Gamma}\mathbf{1}_B([y])L(y)^{-1} \mathscr D_m^-(y)\,\,\mathrm{d}_l^\times y. \end{split} \tag{61}\]

The local Weyl calculation

The second expression in [eq:reflection-theta-average] is now the starting point. At \(S\), \(wa(y)=a(y^{-1})w\) replaces the vector by a fixed rotation and the argument by \(y^{-1}\). Its constant term has already been removed in [eq:reflection-angular-cancellation]. At an exterior place with zero or no digit, \(w\) is compact. We compute the effect of each nonzero digit.

For a row \(\mathfrak m=\mathfrak h\mathfrak d\), call a prime active when its digit is nonzero. Every prime of \(\mathfrak d\) is active, because there \(j_{\mathfrak p}=1\). For \(\mathfrak p\mid\mathfrak h\) put \(j_{\mathfrak p}=v_{\mathfrak p}(\mathfrak h)\bmod6\). The exact set of possible active products in \(\mathfrak h\) is \[ \mathcal F(\mathfrak h)= \left\{\mathfrak f\ \mathrm{squarefree}: \mathfrak f\mid\mathop{\mathrm{rad}}(\mathfrak h),\ \mathfrak p\mid\mathfrak f\ \text{if }\ \mathfrak p\mid\mathfrak h,\ j_{\mathfrak p}\ne0\right\}. \tag{69}\] Only a prime with \(j_{\mathfrak p}=0\) can be inactive. For \(\mathfrak f\in\mathcal F(\mathfrak h)\), its exact inactive scalar is \[I_{\mathfrak h,\mathfrak f} =\prod_{\substack{\mathfrak p\mid\mathfrak h\\ \mathfrak p\nmid\mathfrak f}} (1-q_{\mathfrak p}^{-1}),\] and powerfulness gives \[ \mathfrak f\mid\mathop{\mathrm{rad}}(\mathfrak h),\qquad q_{\mathfrak f}^2\le q_{\mathfrak h}. \tag{70}\] Let \(\mathcal A(\mathfrak f)\) be the integral exterior ideals \(\mathfrak a'_{\mathfrak f}\) supported on \(\mathfrak f\), with every exponent congruent to \(0\) or \(1\) modulo \(3\); set \(\mathcal A(1)=\{1\}\). This is a convenient common support: the actual coefficient \(s_{\mathfrak p,j_{\mathfrak p},e}\) remains zero in the locally vanishing cases of 19.

The local support shows that every frequency with a nonzero grouped exterior coefficient occurs among \[ \nu=\varepsilon\frac{a'_{\mathfrak f}c'(n')^3}{(df)^2}, \qquad \mathfrak a'=\mathfrak a'_{\mathfrak f}\mathfrak c'(\mathfrak n')^3, \qquad \varepsilon\in U_S, \tag{71}\] where \(\mathfrak a'_{\mathfrak f}\in\mathcal A(\mathfrak f)\), \(\mathfrak c'\) is squarefree, and \((\mathfrak c'\mathfrak n',\mathfrak d\mathfrak f)=1\). The displayed elements \(d,f,a'_{\mathfrak f},c',n'\) are allowed generators. The ideals \(\mathfrak c'\) and \(\mathfrak n'\) may share primes, and primes of \(\mathfrak h\) omitted from \(\mathfrak f\) may occur in either. The valuation-one formula forces numerator exponent zero at each prime of \(\mathfrak d\). Away from \(\mathfrak d\mathfrak f\), the valuations \(3k\) and \(3k+1\) determine \(\mathfrak c'\) and \(\mathfrak n'\) uniquely. The remaining ambiguity is the displayed \(S\)-unit, since allowed product generators are unique modulo \(\Gamma\).

For later absolute estimates define \[ \beta_{\mathfrak p}(e)= \begin{cases} 1,&e=0,\\ q_{\mathfrak p}^{-k},&e=1+3k,\ k\ge0,\\ q_{\mathfrak p}^{1/2-k},&e=3k,\ k\ge1,\\ 0,&\text{otherwise}, \end{cases} \qquad \beta_{\mathfrak f}(\mathfrak a'_{\mathfrak f}) =\prod_{\mathfrak p\mid\mathfrak f} \beta_{\mathfrak p}(v_{\mathfrak p}(\mathfrak a'_{\mathfrak f})). \tag{72}\] The local lemma and the spherical magnitudes give \(|s_{\mathfrak p,j,e}|\le\beta_{\mathfrak p}(e)\). The other exterior magnitudes together are at most \(q_{\mathfrak c'}^{-1/2}q_{\mathfrak n'}^{-1}\).

The full reflected expansion

We now assemble these local factors. The \(S\)-integrals form a fixed family of one-variable profiles. Let \(f_{w,\upsilon}(v)\) be the \(S\)-Whittaker function of the chosen vector after the fixed Weyl rotation. Choose once a presentation \[f_{w,\upsilon}(v) =\sum_{\ell=1}^{J_\upsilon}\prod_{a\in S} f_{\upsilon,\ell,a}(v_a).\] The finite \(S\)-functional and the fixed cutoff vector provide such presentations by [prop:theta-input,eq:theta-chosen-S-function]; their lengths are bounded over the finite row-class family. All fixed scalar coefficients are included in the displayed local factors. Write \(H_S\) also for its induced pairing on \(\mathcal V\). For \(\kappa\in\mathcal V\), define \[ \begin{split} H_{\upsilon,\kappa,\ell}(x) &=\int_{X_x}\mathbf{1}_B(\kappa[v]^{-1})H_S(\kappa,[v])^2L_\upsilon(v)\\ &\hspace{25mm}\times \prod_{a\in S}f_{\upsilon,\ell,a}(v_a)\,\,\mathrm{d}_x^\times v,\\ \Phi_{\upsilon,\kappa,\ell}(x) &=\int_0^\infty\mathcal K(T)H_{\upsilon,\kappa,\ell}(x/T) \,\frac{\,\mathrm{d}T}{T},\\ \Phi_{\upsilon,\kappa}(x) &=\sum_{\ell=1}^{J_\upsilon}\Phi_{\upsilon,\kappa,\ell}(x). \end{split} \tag{73}\] The class multiplier is bounded and depends only on \(\mathcal V\), by [eq:reflection-S-cocycle]. The local bounds for the fixed rotated vectors and [lem:sheet-profile,lem:gaussian-smoothing] therefore give one \(\alpha>0\) such that, for every \(N>0\) and integer \(j\ge0\), \[ |(x\partial_x)^j\Phi_{\upsilon,\kappa}(x)| \ll_{N,j}\min(x^\alpha,x^{-N})\qquad(x>0). \tag{74}\] The constants are uniform in \(\upsilon\in U_6\) and \(\kappa\in\mathcal V\). This is a fixed finite family: the functional and tensor terms have been combined in \(\Phi_{\upsilon,\kappa}\), while the bounded class function stays inside its integral. In particular those fixed coefficients will not enter the local majorant \(\beta_{\mathfrak f}\).

Choose representatives \(\varepsilon_u\) for the separate frequency-unit classes \(u\in U_6\). For \[\mathfrak t=(\mathfrak h,\mathfrak f,\mathfrak a'_{\mathfrak f}, \mathfrak n',u),\qquad \mathfrak f\in\mathcal F(\mathfrak h),\quad \mathfrak a'_{\mathfrak f}\in\mathcal A(\mathfrak f),\quad (\mathfrak n',\mathfrak f)=1,\] and class slices \(s_d,s_c\in B\), put \[ \kappa_6(\mathfrak t;s_d,s_c) =[\varepsilon_u][a'_{\mathfrak f}]s_c[n']^3 s_d^{-2}[f]^{-2}\in\mathcal V. \tag{75}\] This is a selected class, rather than a further summation index. For \(\mathfrak p\mid\mathfrak f\), abbreviate \[e_{\mathfrak p}=v_{\mathfrak p}(\mathfrak a'_{\mathfrak f}),\qquad s_{\mathfrak p}=s_{\mathfrak p,j_{\mathfrak p},e_{\mathfrak p}},\qquad b_{\mathfrak p}=b_{j_{\mathfrak p},e_{\mathfrak p}}.\] With \(\upsilon,\mathfrak t,s_d,s_c\) fixed, the next three factors organize the dependence on the two varying ideals: \(\omega\) depends on neither, \(\rho\) depends on the row \(\mathfrak d\) alone, and \(\lambda\) depends on the column \(\mathfrak c'\) alone. All three are independent of \(Z\). The quadratic symbol and norm profile will contain the remaining joint dependence on \(\mathfrak d\) and \(\mathfrak c'\). For \(L=L_\upsilon\) and \(\varepsilon=\varepsilon_u\), define \[ \begin{split} \omega_{\upsilon,\mathfrak t} ={}&I_{\mathfrak h,\mathfrak f} L(a'_{\mathfrak f})^{-1}L(n')^{-3}L(f)^2\\ &{}\times\prod_{\mathfrak p\mid\mathfrak f} s_{\mathfrak p} \chi_{\mathfrak p}\!\left( \varepsilon\frac{a'_{\mathfrak f}}{p^{e_{\mathfrak p}}} (n')^3(f/p)^{-2}\right)^{b_{\mathfrak p}}. \end{split} \tag{76}\] For a squarefree \(\mathfrak d\) with \((\mathfrak d,\mathfrak h\mathfrak n')=1\) and \([d]=s_d\), put \[ \begin{split} \rho_{\upsilon,\mathfrak t,s_d}(\mathfrak d) ={}&L(d)^2\\ &{}\times\prod_{\mathfrak p\mid\mathfrak d} \omega_{\mathfrak p} \chi_{\mathfrak p}\!\left( \varepsilon a'_{\mathfrak f}(n')^3(d/p)^{-2}f^{-2}\right)\\ &{}\times\prod_{\mathfrak p\mid\mathfrak f} \chi_{\mathfrak p}(d)^{-2b_{\mathfrak p}}, \end{split} \tag{77}\] and put it equal to zero otherwise. For a squarefree \(\mathfrak c'\) with \((\mathfrak c',\mathfrak f)=1\) and \([c']=s_c\), put \[ \begin{split} \lambda_{\upsilon,\mathfrak t,s_c}(\mathfrak c') ={}&L(c')^{-1} \prod_{\mathfrak p\mid\mathfrak f} \chi_{\mathfrak p}(c')^{b_{\mathfrak p}}\\ &{}\times\prod_{\mathfrak q\mid\mathfrak c'} g_{2,\mathfrak q} \chi_{\mathfrak q}\!\left( \varepsilon a'_{\mathfrak f}(c'/q)f^{-2}\right)^2, \end{split} \tag{78}\] and put it equal to zero otherwise. All character arguments in these formulas are units at their primes on the stated supports. Thus negative character powers are well-defined. These definitions give one coherent row function and one coherent column function for each frozen tuple and class slice.

The case \(m=0\) remains the separate identity \(B_0(Z)=P_G(1/Z)\) from the coefficient-family section.

The expansion also displays the length used in the second moment. On dyads \(P\le q_{\mathfrak h}<2P\) and \(D\le q_{\mathfrak d}<2D\), put \[ \begin{aligned} Q&=q_{\mathfrak a'_{\mathfrak f}}q_{\mathfrak c'}q_{\mathfrak n'}^3, &\qquad t_{\mathfrak d}&=\frac{q_{\mathfrak d}^2q_{\mathfrak f}^2}{Z},\\ \mathcal L&=\frac{8D^2P}{Z}, & R_0&=Z^\delta\mathcal L, \end{aligned} \tag{88}\] where \(\delta>0\) will be chosen small. By [eq:reflection-active-norm], \(t_{\mathfrak d}\le\mathcal L\). We shall retain \(Q\le R_0\), which in particular gives \(q_{\mathfrak c'}\le R_0\). If \(R_0<1\), this range is empty because \(Q\) is the norm of an integral ideal. The tail estimate below justifies this truncation for every dyad, including that case.

The quadratic large sieve and the moment estimate

We use the quadratic large sieve established over \(\mathbb{Q}\) by Heath-Brown and extended to Hecke families by Goldmakher–Louvel [12, 8]. We state the latter result in the form needed for [eq:reflection-global-expansion] and verify its family hypotheses for the present generator convention.

Weights and the estimate from reflection

Choose a nonzero nonnegative function \(W_1\in C_c^\infty((0,\infty))\), and set \[\widehat W_1(w)=\int_0^\infty W_1(t)t^w\,\frac{dt}{t}.\] Choose a tensor Bruhat–Schwartz function \(W_0\) on \(F_S\), compactly supported in \(F_S^\times\), such that \[ \widehat W_0(0)=0,\qquad M(1/6)\ne0,\qquad M(z)=\int_{F_S^\times}\widehat W_0(y)|y|_S^z\,d^\times y . \tag{98}\] The condition \(\widehat W_0(0)=0\) will remove the zero additive frequency after Poisson summation. The condition \(M(1/6)\ne0\) will ensure that the weight does not kill the residue at \(z=1/6\) in that expansion. These are separate from the support condition on \(W_0\), which removes the original row \(m=0\) from the probe. Here the Fourier sign is the one fixed in 3. We explain the choice and the analytic properties that will be needed. At each place the local zeta functional equation gives, for \(0<s<1\), \[\int_{F_v^\times}\widehat W_v(y)|y|_v^s\,d^\times y =c_v(s)\int_{F_v^\times}W_v(y)|y|_v^{1-s}\,d^\times y, \qquad c_v(1/6)\ne0;\] the nonzero constant incorporates the chosen local measures and Fourier sign [27]. At one complex place take a smooth annular function whose additive integral is zero but whose pairing with \(|y|_v^{5/6}d^\times y\) is nonzero. Such a function is obtained by subtracting suitable multiples of two smooth bumps supported in different radial subannuli: the additive integral is a constant multiple of the pairing with \(|y|_v d^\times y\), and the two radial weights are not proportional. At all other places take nonnegative nonzero annular functions. The local identity then proves both requirements in [eq:poisson-Fourier-weight].

For every \(\sigma>0\), each local integral of \(|\widehat W_v(y)||y|_v^\sigma\) is finite. At a finite place this follows by summing the geometric series of shells near zero and using compact support at infinity; at a complex place it follows from boundedness at zero and Schwartz decay at infinity. The same assertions hold after any number of derivatives in a complex logarithmic norm. Thus \(M\) is holomorphic for \(\operatorname{Re}z>0\), and integration by parts in that norm gives, for every compact interval \(J\subset(0,\infty)\) and every \(A>0\), \[ |M(\sigma+i\tau)|\ll_{J,A}(1+|\tau|)^{-A} \quad(\sigma\in J). \tag{99}\] The analogous estimate holds for \(\widehat W_1\) on every fixed real strip.

For an exterior ideal \(\mathfrak s\), with an allowed generator \(s\), put \[g_{\mathfrak s}(k) =\sum_{d\bmod\mathfrak s}\chi_{\mathfrak s}(d)\mathrm{e}(kd/s).\] The value is independent of the allowed generator: a change by \(\epsilon^6\), \(\epsilon\in U_S\), is absorbed by the same change of the residue variable. Use product multiplicative Haar measures for which the norm decomposition of \(F_S^\times\) has the form \[d^\times y=\frac{dT}{T}\,dt,\qquad y=t\ell_T,\quad t\in F_S^1.\] The measure on \(F_S^1/U_S^6\) below is the quotient measure induced by \(dt\). This choice also fixes the normalization in \(M\). Define \[ \begin{split} I(X,Y,Z)=\int_{F_S^1/U_S^6} \frac1Y\sum_{\mathfrak s}W_1(q_{\mathfrak s}/Y) \frac1{\sqrt{q_{\mathfrak s}X}}\sum_{m\in R} q_{\mathfrak s}^{-1/2}g_{\mathfrak s}(-m) W_0\!\left(\frac{m}{t_0\ell_{q_{\mathfrak s}X}}\right) B_m(Z)\,dt_0 . \end{split} \tag{100}\] All undecorated ideal sums in this section are over nonzero integral exterior ideals. The summand for \(m=0\) vanishes because \(W_0\) is supported in \(F_S^\times\). For \(\epsilon\in U_S^6\), \(\chi_{\mathfrak A}(\epsilon)=\chi_{\mathfrak s}(\epsilon)=1\); changing \(m\) to \(\epsilon m\) shows that the integrand descends to the quotient. That quotient is compact by 9, and we use representatives in a fixed bounded subset of \(F_S^1\). For each such representative the \(m\)-sum has finite support. The Gaussian weight in \(B_m\) makes its ideal sum absolutely convergent.

We first record the additive estimate in the form also used later for conductor characters. If \(\mathfrak s=R\), its one residue class is understood to give the single fraction \(0\).

Poisson summation and three Mellin variables

For an exterior ideal \(\mathfrak A\) with allowed generator \(A\), write \[g_{\chi_{\mathfrak A}}(A,k) =\sum_{v\bmod\mathfrak A}\chi_{\mathfrak A}(v)\mathrm{e}(kv/A)\] and, for \(h\in R\), define \[ F(s,A,h)= \sum_{\substack{d\bmod\mathfrak s\\h\equiv Ad\pmod{\mathfrak s}}} \chi_{\mathfrak s}(d)\, g_{\chi_{\mathfrak A}}\!\left(A,\frac{h-Ad}{s}\right). \tag{104}\] The argument of the last Gauss sum is in \(R\) by the congruence. The definitions for the unit ideal mean a single residue and the value \(1\) for its character.

The finite Fourier transform of the row coefficient modulo \(\mathfrak s\mathfrak A=sAR\) is exactly \[ \sum_{m\bmod\mathfrak s\mathfrak A} q_{\mathfrak s}^{-1/2}g_{\mathfrak s}(-m) \chi_{\mathfrak A}(m)\mathrm{e}(hm/(sA)) =q_{\mathfrak s}^{1/2}F(s,A,h). \tag{105}\] In fact expand \(g_{\mathfrak s}\), write \(m=v+Aj\) with \(v\bmod\mathfrak A\) and \(j\bmod\mathfrak s\), and sum over \(j\). The sum of \(\mathrm{e}((h-Ad)j/s)\) is \(q_{\mathfrak s}\) precisely when \(h\equiv Ad\pmod{\mathfrak s}\), and is zero otherwise. The remaining sum over \(v\) is the one in [eq:finite-Poisson-factor]. This argument does not require \(\mathfrak s\) and \(\mathfrak A\) to be coprime.

The dilation \(t_0\ell_{q_{\mathfrak s}X}\) has additive modulus \(q_{\mathfrak s}X\), and the lattice \(sAR\) has index \(q_{\mathfrak s}q_{\mathfrak A}\). Apply Poisson summation to the \(m\)-sum and use [eq:finite-Poisson-transform]. Its dilation divided by the index is \(X/q_{\mathfrak A}\); after the two square root factors in the probe the multiplier is \[(q_{\mathfrak s}X)^{-1/2} \frac X{q_{\mathfrak A}}q_{\mathfrak s}^{1/2} =\frac{X^{1/2}}{q_{\mathfrak A}}.\] The Fourier argument at \(h\) is \[t_0\ell_{q_{\mathfrak s}X}\frac h{sA},\qquad \left|\ell_{q_{\mathfrak s}X}\frac h{sA}\right|_S =\frac{Xq_h}{q_{\mathfrak A}}.\] These norm factors will determine all three Mellin exponents.

For \(\epsilon\in U_S^6\), changing \(d\) to \(\epsilon d\) and the Gauss residue variable \(v\) to \(\epsilon^{-1}v\) in [eq:finite-Poisson-factor] shows \(F(s,A,\epsilon h)=F(s,A,h)\). Sum the nonzero \(h\)’s in such orbits and unfold the quotient \(F_S^1/U_S^6\). If \[V_0(T)=\int_{F_S^1}\widehat W_0(t\ell_T)\,dt,\] the orbit of \(h\) contributes \(F(s,A,h)V_0(Xq_h/q_{\mathfrak A})\). Multiplication by the norm-one part of \(\ell_{q_{\mathfrak s}X}h/(sA)\) preserves the sheet measure, so no shape factor remains. This unfolding is absolutely justified: for each \(\sigma>0\), the local integrability used above gives an absolute sheet majorant \(O_\sigma(T^{-\sigma})\), by using one complex weighted supremum and the other weighted local integrals. Taking \(\sigma>1\) sums the \(q_u^{-\sigma}q_{\mathfrak r}^{-6\sigma}\) majorant, while the Gaussian handles the ideal \(\mathfrak A\); this permits Tonelli’s theorem in the orbit sum. The zero frequency contributes nothing because \(\widehat W_0(0)=0\).

By the measure disintegration, the Mellin transform of \(V_0\) is \(M\). The same weighted derivative majorants in the complex norm variables justify differentiation of the sheet integral in \(\log T\), so \(V_0(T)\) is smooth for \(T>0\). Its absolute weighted integrability and [eq:poisson-M-decay] then give pointwise Mellin inversion by Fourier inversion in \(\log T\) on each line \(\operatorname{Re}z>0\). By [eq:frequency-decomposition], write the orbits uniquely as \(h=ur^6\), where \(u\in\mathcal U\) and \(\mathfrak r\) is an exterior ideal with allowed generator \(r\). The principal row \(u=1\) consists of the surviving nonzero sixth-power frequencies \(h=r^6\). Hence the precise Poisson expression is \[ \begin{split} I(X,Y,Z) ={}&\frac{X^{1/2}}Y \sum_{\substack{\mathfrak c\ {\rm squarefree}\\ \mathfrak n,\mathfrak s}} \frac{\gamma_2(\mathfrak c)\vartheta(\mathfrak A)} {q_{\mathfrak c}^{1/2}q_{\mathfrak n}q_{\mathfrak A}}\, P_G(q_{\mathfrak A}/Z)W_1(q_{\mathfrak s}/Y)\\ &{}\times \sum_{u\in\mathcal U}\sum_{\mathfrak r} F(s,A,ur^6) V_0\!\left(\frac{Xq_uq_{\mathfrak r}^6}{q_{\mathfrak A}}\right), \qquad \mathfrak A=\mathfrak c\mathfrak n^3. \end{split} \tag{106}\] Here \(P_G\) is the Gaussian of [eq:reflection-gaussian].

Use Mellin inversion for \(P_G,W_1,V_0\), with transforms \(e^{t^2},\widehat W_1(w),M(z)\), respectively. The factors involving \(\mathfrak A\) are \(q_{\mathfrak A}^{-1}q_{\mathfrak A}^{-t}q_{\mathfrak A}^{z}\). Set \(t=\xi+z-1\). We obtain \[ \begin{split} I(X,Y,Z)=\frac1{(2\pi i)^3} \int_{(4)}\int_{(4)}\int_{(2)} &X^{1/2-z}Z^{\xi+z-1}Y^{w-1} e^{(\xi+z-1)^2}M(z)\widehat W_1(w)\\ &{}\times\sum_{u\in\mathcal U}q_u^{-z} \mathcal F_u(\xi,w,z)\,dz\,dw\,d\xi , \end{split} \tag{107}\] where the innermost line is \(\operatorname{Re}z=2\), the middle line is \(\operatorname{Re}w=4\), and the outer line is \(\operatorname{Re}\xi=4\), and \[ \mathcal F_u(\xi,w,z)= \sum_{\substack{\mathfrak c\ {\rm squarefree}\\ \mathfrak n,\mathfrak s,\mathfrak r}} \gamma_2(\mathfrak c)\vartheta(\mathfrak A)F(s,A,ur^6) q_{\mathfrak c}^{-1/2-\xi}q_{\mathfrak n}^{-1-3\xi} q_{\mathfrak s}^{-w}q_{\mathfrak r}^{-6z}, \qquad \mathfrak A=\mathfrak c\mathfrak n^3. \tag{108}\] All interchanges used for [eq:poisson-mellin] are in absolute convergence. For example, \(|F(s,A,h)|\le q_{\mathfrak s}q_{\mathfrak A}\), \(|\gamma_2(\mathfrak c)|=1\), and the ideal count, including the finitely many S-unit classes for \(u\), give absolute convergence on these lines. The factors \(M(z)\) and \(\widehat W_1(w)\) have arbitrary vertical polynomial decay, while the Gaussian has Gaussian decay in \(\operatorname{Im}(\xi+z)\). Changing height variables to \((\operatorname{Im}(\xi+z),\operatorname{Im}z,\operatorname{Im}w)\) proves absolute integrability as well.

The local correction

We extend the notation \(L^S(s,\eta)\) from 4 to a Hecke character on exterior primes: the factors at \(S\) are omitted, and any additional zero prime values are retained in the Euler product. Thus, for example, \[L^S(w,\chi_\bullet(u)) =\prod_{\mathfrak p\notin S} (1-\chi_{\mathfrak p}(u)q_{\mathfrak p}^{-w})^{-1} \quad(\operatorname{Re}w>1).\] As before, \(\zeta_F^S(s)=\prod_{\mathfrak p\notin S}(1-q_{\mathfrak p}^{-s})^{-1}\).

The two regions serve different contour movements in the next section. The first permits the small and intermediate nonprincipal rows to place \(w\) at \(.003\) and \(z\) at \(.4\), while \(\xi\) can reach \(.998\) once the reciprocal denominator has been controlled. The second contains the principal numerator poles \(w=1\), \(z=1/6\) and the nearby lines \(w=.95\), \(z=.16\) used to isolate their double residue. The nowhere-zero conclusion for \(u=1\) will ensure that its correction does not cancel a singularity of \(1/L^S(\xi,\eta)\).

For the remainder of the proof, write \[\mathcal H_\eta(s):=\mathcal H_1(s,1,1/6).\] This is the principal specialization announced in 4. Since \((w,z)=(1,1/6)\) lies in the second region of [eq:euler-regions], \(\mathcal H_\eta\) is holomorphic, bounded, and nowhere zero on an open neighborhood of the closed half-plane \(\operatorname{Re}s\ge .998\), uniformly in height.

A conductor large sieve and a zero detector

Let \(\omega_u\) denote the primitive Hecke character inducing \(\eta\overline{\chi_\bullet(u)}\). By 10, on a dyad \(U\le q_u<2U\), \[ \mathrm{N}\mathfrak f_{\omega_u}\ll_{F,S,\eta}U, \qquad \#\{u\in\mathcal U:U\le q_u<2U\}\ll_{F,S}U . \tag{120}\] The second bound follows from the ideal count and the finite number of S-unit classes modulo sixth powers. The characters \(\omega_u\) in this family are distinct: equality cancels the fixed twist and forces \(u/v\) to be a sixth power in \(F\); the selected exterior valuations then agree, and the quotient is in \(U_S^6\). Their local components at \(S\) run through a fixed finite set, since the Kummer part depends on \(F_v^\times/F_v^{\times6}\). The principal numerator occurs only for \(u=1\). The primitive denominator \(\omega_u\) may be principal for a different row, so it will be treated separately in zero detection.

We give the conductor estimate required for that detection. In the next lemma and in the detector, ideals without the adjective exterior are integral ideals of \(\mathcal O_F\), and \(q_{\mathfrak n}=\mathrm{N}\mathfrak n\) also for those ideals.

We use standard Hecke continuation and the functional equation in a deliberately conservative convexity bound. For a nonprincipal primitive finite-order \(\omega\) with \(\mathrm{N}\mathfrak f_\omega\ll U\), there is an \(A_F>0\), depending on the fixed field, such that on any of the fixed strips below, in particular for \(.003\le\sigma\le4\), \[ |L_F(\sigma+it,\omega)| \ll_{F,S,\eta,\epsilon}U^{1/2+\epsilon}(2+|t|)^{A_F}. \tag{122}\] This follows from the functional equation, the absolutely convergent Euler product on a right line, and the Phragmén–Lindelöf principle; the gamma factors contribute the field-dependent height power [27]. The same estimates give polynomial height bounds on any fixed strip for \((s-1)\zeta_F(s)\). Omitting factors at a fixed set or at primes dividing \(u\) changes a positive-real-part estimate by \(q_u^\epsilon\): for a fixed \(\sigma_*>0\), each of \[\prod_{\mathfrak p\mid(u)R}(1+q_{\mathfrak p}^{-\sigma_*}), \qquad \prod_{\mathfrak p\mid(u)R}(1-q_{\mathfrak p}^{-\sigma_*})^{-1}\] is \(O_{\epsilon,\sigma_*}(q_u^\epsilon)\), by bounding a sufficiently large prime’s factor by \(q_{\mathfrak p}^\epsilon\) and absorbing the finitely many other primes into the constant.

The principal component and the other frequency rows

Use the scales in [eq:reflection-scales], and let \(\sigma_\xi,\sigma_w,\sigma_z\) denote the real parts of \(\xi,w,z\). Write \(E_*(\sigma_\xi,\sigma_w,\sigma_z)\) for the real power of \(Z\) outside the Dirichlet factors in [eq:poisson-mellin]. In the principal row, taking the residues at \(w=1\) and \(z=1/6\) leaves the exponent \(C_*(\xi)\) in the second line: \[ \begin{split} E_*(\sigma_\xi,\sigma_w,\sigma_z) &=\frac a2+\sigma_\xi-1+h_0\sigma_z+b(\sigma_w-1),\\ C_*(\xi)&=\frac a2+\xi-1+\frac{h_0}{6} =\xi-\frac8{15},\qquad C_*(1)=\frac7{15}. \end{split} \tag{129}\] Thus \(C_*(1)=7/15\) is the reference power for comparing the principal remainders and the other frequency rows.

We specify a decay observation that justifies the contour limits and, later, the truncated shifts. On any fixed real strips used below, for every \(A>0\), \[ \left|e^{(\xi+z-1)^2}M(z)\widehat W_1(w)\right| \ll_A e^{-(\operatorname{Im}\xi+\operatorname{Im}z)^2} (1+|\operatorname{Im}z|)^{-A}(1+|\operatorname{Im}w|)^{-A}. \tag{130}\] The constants can depend on those strips. Away from the poles extracted below, all numerator Hecke and zeta functions on them have polynomial height growth, and \(\mathcal H_u\) is uniformly bounded by \(q_u^\epsilon\). After increasing \(A\), their height powers are absorbed in [eq:triple-height-decay]. In particular for every \(A_1>0\), \[\int_{\mathbb{R}}e^{-(T+v)^2}(1+|v|)^{-A}\,dv \ll_{A_1}(1+|T|)^{-A_1}\] with \(A\) sufficiently large. Thus integration in \(z,w\) gives arbitrary polynomial decay in the remaining \(\xi\)-height as well. On lines \(\operatorname{Re}\xi>1\) the reciprocal denominator is globally bounded by its absolutely convergent Euler product. These facts justify all infinite contour shifts on such lines by their horizontal limits.

Let \(I_1(Z)\) denote the contribution of \(u=1\) to the initially absolutely convergent formula [eq:poisson-mellin], on the scales \((X,Y)=(Z^a,Z^b)\). Its numerator is \(\zeta_F^S(w)\). Put \(\delta_1=10^{-7}\), \(\sigma_1=1+\delta_1\), and \(\zeta_1=1/6+\delta_1\). Move the initial lines to \[\operatorname{Re}\xi=\sigma_1,\qquad \operatorname{Re}w=\sigma_1,\qquad \operatorname{Re}z=\zeta_1.\] These moves cross no poles: both zeta arguments remain to the right of one, and the reciprocal denominator is in absolute convergence. The second region of [eq:euler-regions] contains all the moves. Now move \(w\) to \(.95\), crossing its simple pole at \(1\), and call the remaining triple integral \(R_w(Z)\). In the \(w=1\) residue move \(z\) to \(.16\), crossing the pole of \(\zeta_F^S(6z)\) at \(1/6\), and call the remaining residue integral \(R_z(Z)\). Their contours are \[\begin{aligned} R_w:&\quad (\operatorname{Re}\xi,\operatorname{Re}w,\operatorname{Re}z)=(\sigma_1,.95,\zeta_1),\\ R_z:&\quad (\operatorname{Re}\xi,w,\operatorname{Re}z)=(\sigma_1,1,.16). \end{aligned}\] The other term is the double residue. With its evaluation \(c_0f_\eta(Z)\) below, the principal component is exactly \[I_1(Z)=c_0f_\eta(Z)+R_w(Z)+R_z(Z).\] The powers of the two remainders relative to \(C_*(1)\) are \[\begin{split} E_*(\sigma_1,.95,\zeta_1)-C_*(1) &=(1+h_0)\delta_1-.05b=-.00664989,\\ E_*(\sigma_1,1,.16)-C_*(1) &=\delta_1+h_0(.16-1/6) =-.000666566666\ldots . \end{split}\] The height bounds just given and 24 therefore give \[ \begin{aligned} R_w(Z)&=O_{F,S,\eta,W_0,W_1}\bigl(Z^{C_*(1)-.0005}\bigr),\\ R_z(Z)&=O_{F,S,\eta,W_0,W_1}\bigl(Z^{C_*(1)-.0005}\bigr). \end{aligned} \tag{131}\]

To evaluate the double residue, write \(R_S=\mathop{\mathrm{Res}}_{s=1}\zeta_F^S(s)>0\). It has the form \(c_0f_\eta(Z)\), where \[ \begin{split} c_0&=\frac{R_S^2}{6}M(1/6)\widehat W_1(1)\ne0,\\ f_\eta(Z)&=\frac1{2\pi i}\int_{(2)} Z^{C_*(\xi)}e^{(\xi-5/6)^2} \frac{\mathcal H_\eta(\xi)}{L^S(\xi,\eta)}\,d\xi . \end{split} \tag{132}\] Indeed the residue of \(\zeta_F^S(6z)\) is \(R_S/6\), and \(\widehat W_1(1)=\int_0^\infty W_1(t)\,dt>0\). The other factor is nonzero by [eq:poisson-Fourier-weight]. The norm measures were fixed before the probe, so this is the exact scalar in its normalization. The line of the double residue was initially \(\operatorname{Re}\xi=\sigma_1\); we moved it right to \(2\) to write [eq:principal-Mellin-integral]. This uses only absolute convergence of \(1/L^S(\xi,\eta)\) for \(\operatorname{Re}\xi>1\).

It remains to bound the rows with \(u\ne1\). Split their sum on the initial lines into dyads \(U\le q_u<2U\), \(U=1,2,4,\ldots\), before shifting any of them. Each dyad contains finitely many rows; the full sum on the initial lines is absolutely convergent. Denote by \(\mathcal J_{\mathcal C}(Z)\) the triple integral of a subset \(\mathcal C\) of such a dyad, and set \[\mathcal J_U^{\mathrm{np}}(Z) =\mathcal J_{\{u\ne1:U\le q_u<2U\}}(Z).\] The numerator for each of these rows is entire. From [eq:Hecke-conservative-convexity] and the omitted-factor bound, \[ L^S(w,\chi_\bullet(u)) \ll_\epsilon U^{1/2+\epsilon}(2+|\operatorname{Im}w|)^{A_F} \qquad(.003\le\operatorname{Re}w\le4). \tag{133}\] On \(\operatorname{Re}\xi=\sigma_1\), independently of \(u\) and the height, \[|L^S(\xi,\eta\overline{\chi_\bullet(u)})^{-1}| \le\prod_{\mathfrak p\notin S}(1+q_{\mathfrak p}^{-\sigma_1}) \le\zeta_F(\sigma_1).\] For each row move the initial lines to \[\operatorname{Re}\xi=\sigma_1,\qquad \operatorname{Re}w=.003,\qquad \operatorname{Re}z=.4.\] One may first move \(\xi\), then \(z\), then \(w\). The \(z\)-line remains above \(1/6\); the numerator is entire; and the denominator stays in absolute convergence. The first region of [eq:euler-regions] contains the whole move. If \(|\mathcal C|\ll U^\alpha\), the bounds above give \[ |\mathcal J_{\mathcal C}(Z)| \ll_\epsilon Z^{E_*(\sigma_1,.003,.4)} U^{\alpha+1/2-.4+\epsilon}. \tag{134}\] The factor \(U^{-.4}\) is the \(q_u^{-z}\) in [eq:poisson-mellin]. The height integral is bounded by [eq:triple-height-decay].

To see how the size of the frequency determines the contours, rewrite [eq:dyadic-contour-bound] as \[|\mathcal J_{\mathcal C}(Z)| \ll_\epsilon Z^{a/2+\sigma_1-1+b(.003-1)} U^{\alpha+1/2+\epsilon} \left(\frac{Z^{h_0}}U\right)^{.4}.\] For a general \(z\)-line, the corresponding factor is \((Z^{h_0}/U)^{\operatorname{Re}z}\). Its dependence on \(\operatorname{Re}z\) changes direction at \(U=Z^{h_0}=Z^{.1}\): below that scale a rightward \(z\)-shift enlarges this factor, while above it such a shift reduces the factor. We retain the current low-\(w\) contour through \(U\le Z^{h_0+.00001}\), and use a rightward \(z\)-shift beyond that enlarged threshold.

The common \(Z\)-exponent on the current contour, relative to the principal scale, is \[E_*(\sigma_1,.003,.4)-C_*(1) =10^{-7}+\frac7{300}-.132601 =-.109267566666\ldots .\] For \(U\le Z^{.09}\), use [eq:dyadic-contour-bound] with \(\alpha=1\). The \(U^{1.1}\) factor adds at most \(.09(1.1)=.099\) to this relative exponent, so these rows are already smaller than the principal scale.

We use 26 only in the remaining interval \[ Z^{.09}<U\le Z^{h_0+.00001}=Z^{.10001}, \tag{135}\] where its threshold \(U_0\) is met for all sufficiently large \(Z\). For the rows in \(\mathcal E(U)\), use the same contour estimate with \(\alpha=1/2\). The \(U^{.6}\) factor then adds at most \(.10001(.6)=.060006\) to the same relative exponent.

For each remaining row truncate its \(\xi\)-line at \(|\operatorname{Im}\xi|\le U^\rho\), and move that finite portion to \(\operatorname{Re}\xi=.998\). The rectangle and its two horizontal sides satisfy [eq:good-reciprocal]; hence the new central integral satisfies \[ \ll_\epsilon Z^{E_*(.998,.003,.4)} U^{1+1/2-.4+\epsilon} \tag{136}\] after summing all the good rows of the dyad. The estimate does not use the reciprocal outside the specified rectangle. Since \(E_*\) is linear in \(\operatorname{Re}\xi\) with coefficient one, this move gains \(Z^{.998-\sigma_1}=Z^{-.0020001}\). Including the \(U^{1.1}\) cost up to the upper endpoint of [eq:intermediate-frequency-range], the relative exponent is at most \[ -.002+\frac7{300}-.132601+.10001(1.1) =-\frac{377}{300000}=-.001256666666\ldots . \tag{137}\]

To bound the discarded and horizontal portions, integrate in \(z,w\) first in [eq:triple-height-decay]. For every prescribed \(A_1>0\) the discarded tails on \(\operatorname{Re}\xi=\sigma_1\) and the horizontal portions at \(\operatorname{Im}\xi=\pm U^\rho\) are, in total over a good dyad, at most \[ \ll_{A_1,\epsilon} Z^{E_*(\sigma_1,.003,.4)} U^{1+1/2-.4+\epsilon-\rho A_1}. \tag{138}\] On the horizontal portions the real exponent is bounded by its value at \(\sigma_1\); the reciprocal is bounded by [eq:good-reciprocal]. On the discarded original tails it is bounded by absolute convergence. The convolution estimate following [eq:triple-height-decay], used with pointwise exponent \(A_1+2\) before integrating the discarded \(\xi\)-tails, supplies \(U^{-\rho A_1}\) in both cases. Since \(U>Z^{.09}\), choosing \(A_1>1/\rho\), after \(\rho\) is fixed, gives more than enough additional saving for these terms.

All these comparisons leave a fixed margin after the arbitrary small power losses. The sums over the \(O(\log Z)\) dyads in these ranges cost only another arbitrarily small power.

Finally consider \(U>Z^{.10001}\). Now \(Z^{h_0}/U\le Z^{-.00001}\), so a fixed rightward \(z\)-shift suppresses the whole dyad. Keep \(\operatorname{Re}\xi=\operatorname{Re}w=4\) and move \(z\) to the fixed right line \(\operatorname{Re}z=R_0=350000\). The move stays in the Euler region, crosses no pole, and uses the holomorphy of \(M\) on \(\operatorname{Re}z>0\). Both Hecke factors are bounded by absolute convergence on these lines, so 24 and [eq:triple-height-decay] give \[|\mathcal J_{\mathcal C}(Z)| \ll_\epsilon Z^{3.849}U^{1+\epsilon} \left(\frac{Z^{.1}}U\right)^{R_0} =Z^{3.849+.1R_0}U^{1-R_0+\epsilon}.\] The dyads sum geometrically. At their lower endpoint the power of \(Z\) is \[ 3.849+.1R_0+.10001(1-R_0+\epsilon) =.44901+.10001\epsilon . \tag{139}\] This is strictly less than \(C_*(1)-.01\) for small \(\epsilon\). The very large \(z\)-line is fixed; its weight constants may be large, but are independent of \(Z\) and \(U\).

Choose the preliminary losses small enough that their total cost in the finite frequency ranges, including the dyadic count, is less than \(10^{-4}\) in the \(Z\)-exponent. This is possible since \(U\le Z^{.10001}\) there. The smallest central saving in [eq:good-numerical-margin] is larger than \(0.0012\); [eq:xi-tail-bound] and [eq:far-frequency-exponent] have larger savings. We have therefore proved \[ \sum_{j\ge0}\mathcal J_{2^j}^{\mathrm{np}}(Z) \ll Z^{C_*(1)-.0004}. \tag{140}\] Each dyadic integral here is obtained from its initial absolutely convergent integral, and the preceding bounds justify their sum.

Completion of the Mellin argument

The numerical exponents established above, before the indicated small losses, are collected in 1. In particular the exponent from reflection is \[\frac{2M_0-1-b}{2}=\frac{933}{2000} =C_*(1)-\frac1{6000}.\]

Combining the principal decomposition \(I_1=c_0f_\eta+R_w+R_z\), [eq:principal-remainders,eq:nonprincipal-row-total], and [eq:poisson-mellin] gives, for sufficiently large \(Z\), \[I(Z^a,Z^b,Z)=c_0f_\eta(Z)+O(Z^{C_*(1)-.0004}).\] Use 23 with, for example, a loss \(10^{-5}\). Since \(1/6000-10^{-5}>1/20000\) and \(c_0\ne0\), the preceding identity implies the safe bound \[ f_\eta(Z)\ll_{F,S,\eta,W_0,W_1}Z^{7/15-1/20000} \qquad(Z\text{ sufficiently large}). \tag{141}\]

Set \[\mathscr A(\xi)=e^{(\xi-5/6)^2} \frac{\mathcal H_\eta(\xi)}{L^S(\xi,\eta)}.\] For \(\operatorname{Re}\xi>1\) this is holomorphic. In the definition of \(f_\eta\) in [eq:principal-Mellin-integral], its \(\xi\)-line may be moved right from \(2\) to any fixed real part \(\sigma_R>2\). The reciprocal there is in absolute convergence, the correction is holomorphic and bounded, and the Gaussian gives rapid vertical decay. It follows that for every \(A_0>0\), \[ f_\eta(Z)=O_{A_0}(Z^{A_0})\qquad(0<Z\le1), \tag{142}\] by taking \(\sigma_R>A_0+8/15\). Thus this endpoint estimate has used no zero exclusion to the left of one.

Consider \[ \mathcal T(s)=\int_0^\infty f_\eta(Z)Z^{-C_*(s)}\,\frac{dZ}{Z}. \tag{143}\] By [eq:f-large-bound,eq:f-small-bound], it converges locally uniformly, and hence defines a holomorphic function, for \[\operatorname{Re}s>1-\frac1{20000}.\] Indeed at infinity the real exponent is \(1-1/20000-\operatorname{Re}s<0\), and at zero [eq:f-small-bound] allows an arbitrarily large exponent. The same majorants with powers of \(\log Z\) justify differentiation under the integral.

We identify this transform on its original line without a formal contour interchange. Put \(t=\log Z\). From [eq:principal-Mellin-integral], \[e^{-C_*(2)t}f_\eta(e^t) =\frac1{2\pi}\int_{\mathbb{R}}e^{i\tau t}\mathscr A(2+i\tau)\,d\tau .\] The function \(\mathscr A(2+i\tau)\) is continuous and rapidly decreasing by the Gaussian and the bounded Euler factors. The endpoint bounds on \(f_\eta\) make the left side integrable in \(t\). Fourier inversion therefore gives \[ \mathcal T(2+i\tau)=\mathscr A(2+i\tau)\qquad(\tau\in\mathbb{R}). \tag{144}\] For \(\operatorname{Re}s>1-1/20000\), the second neighborhood in 24 makes the numerator \[e^{(s-5/6)^2}\mathcal H_\eta(s)\] holomorphic and nowhere zero. The Hecke continuation makes \(\mathscr A(s)\) meromorphic there. The identity theorem, first on \(\operatorname{Re}s>1\) using [eq:Mellin-identification] and then for meromorphic functions on the connected larger half-plane, identifies \(\mathscr A\) with the holomorphic function \(\mathcal T\). A zero of \(L^S(s,\eta)\) in that half-plane would be a pole of \(\mathscr A\), since its numerator is nonzero, which is impossible. A pole of the principal \(L\)-function simply makes its reciprocal vanish.

The finite factors removed from \(L_F(s,\eta)\) are either one or \(1-\eta(\mathfrak p)q_{\mathfrak p}^{-s}\) with \(|\eta(\mathfrak p)|=1\); their zeros have real part zero. Thus the same zero exclusion holds for the full \(L_F(s,\eta)\). The obtained half-plane contains \(\operatorname{Re}s>1-10^{-6}\). All onsets and constants in the proof were allowed to depend on \(F,S,\eta\), but the numerical saving \(1/20000\) is absolute. A field-dependent onset in [eq:f-large-bound] affects only a finite portion of the integral and does not change its half-plane of convergence. Since the initial field and finite-order character were arbitrary in the scope of 2, this proves that theorem.

Field bounds and descent of the zero-free region

The field \(K_q\) need not be cyclotomic. We first place it in an abelian extension of a cyclotomic field, so that Theorem 2 applies, and then descend the resulting zero-free region to \(\zeta_{K_q}\).

The width in (146) is independent of \(q\). It remains to turn it into a prime-ideal estimate whose constants retain only the explicit dependence on degree and discriminant in (145).

A smooth prime-ideal estimate

Smoothing permits an absolutely convergent sum over zeros. The following form of the explicit formula is useful because its error is linear in \(\log D+n\). We give the calculation with this dependence explicit.

A block sieve with bounded coefficients

The advantage of the following form of the Brun–Hooley sieve is that its remainders occur without divisor weights. Its underlying product inequalities are those of Ford and Halberstam [6].

Rough-number densities

Recall the density \(D_\gamma(s)\) from (164). Its \(j\)th term describes a product of \(j\) primes whose logarithms, in units of \(\log x\), sum to \(s\) and are at least \(\gamma\); the factor \(1/j!\) normalizes the ordered prime lists. These are Buchstab’s rough-number densities, expressed in logarithmic coordinates. The identity below is the corresponding least-prime-factor decomposition [4]. The normalization and regularity were established in Section 10; we now prove the two estimates needed to subtract composite mass.

For the product \(V(y)\) in (154), Mertens’ theorem gives \[ V(y)\sim\frac{\exp(-\gamma_E)}{\log y}, \tag{177}\] where \(\gamma_E\) is Euler’s constant.

The term \(1\) in (178) is the one-prime contribution. The integral accounts for the composite contributions by selecting their least prime factor. This is the cancellation that will leave a positive prime mass in the construction below.

The weights and the Type II estimate

Use the parameters \(L,W,V\), the prime groups \(\mathcal P_i\), and the marks \(\mathcal W\) of (154), (155), and (157), with \(q=1/2\). All prime-indexed sums and products below are over rational primes. The integer \(K\ge1\) and exponents \(0.1<a_1<\cdots<a_K<0.2\) are fixed before \(x\) grows; their eventual choice will follow the other sieve parameters. The groups are disjoint for large \(x\), and Mertens’ theorem gives \(V_i=\log2+o(1)\). Their primes are then coprime to \(M\) and lie in \((\exp(L^{0.1}),\exp(L^{0.3}))\).

The factor \(\omega_i(h)\) counts choices of a prime divisor of \(h\) from the \(i\)th group. Thus \(\mathcal W(h)\) vanishes unless every group contributes, while \(\omega_i(h)2^{1-\omega_i(h)}\) remains bounded when several primes from one group divide \(h\). These choices are the marks in the Type II estimate below.

Call a positive integer a group integer if all its prime divisors belong to \(\bigcup_i\mathcal P_i\), and include \(1\). Use the complete group-prime part \(h_{\mathcal P}\) defined in (156). Fix a smooth function \(\Psi\) compactly supported in \((1,2)\), with \(0\le\Psi\le1\) and \(\int\Psi>0\), and define \[ \begin{split} F(h)&=\mathbf{1}_{\{h/h_{\mathcal P}=cQ\text{ for a prime }Q>x^{0.9}\}},\\ w(d)&=\mathbf{1}_{\{d\equiv u\pmod M\}}\Psi(d/x) F(d-1)\mathcal W(d-1)\qquad(d\ge2). \end{split} \tag{182}\] Set \(w(1)=0\). Since \((ph)_{\mathcal P}=p h_{\mathcal P}\) for every group prime \(p\), including when \(p\mid h\), we have \[ F(ph)=F(h)\qquad(h\ge1,\ p\in\textstyle\bigcup_i\mathcal P_i). \tag{183}\] On the support of \(w\), the representation \(d=crQ+1\) is unique: \(r=(d-1)_{\mathcal P}\), since neither \(c\) nor \(Q\) has a group prime factor. In particular, \[ r\le x^{0.11},\qquad \mathcal W(d-1)=\mathcal W(r),\qquad 0\le w(d)\le B_K \tag{184}\] for a constant \(B_K\). The last assertion follows from \(j2^{1-j}\le1\) for each integer \(j\ge1\); in fact \(0\le\mathcal W(h)\le B_K\) for every \(h\ge1\).

Write \(X_0=\sum_d w(d)\) for the total mass. Our aim is to prove that the prime mass is \(\gg X_0/L\), while \(X_0\gg x/L\). The bound \(w\le B_K\) will then give the required count of primes. The marked estimate supplies the comparison needed to subtract composite mass.

Use the rough-number density \(D_\gamma(s)\) from (164). For fixed \(\gamma>0\) and integers \(m>x^\gamma\), put \[ R_\gamma(m)=\mathbf{1}_{\{P^-(m)>x^\gamma\}},\qquad B_\gamma(m)=\frac{D_\gamma(\log m/L)}{LV(W)} \mathbf{1}_{\{P^-(m)>W\}}. \tag{185}\] We will replace \(R_\gamma\) by \(B_\gamma\) inside sums weighted by \(w(mn)\): the factor \(D_\gamma/(LV(W))\) compensates for retaining only \(W\)-roughness. Lemma 30 and the character-transform bound proved in Proposition 31 permit this replacement after the fixed congruence is expanded in characters. We verify that application next. All Dirichlet characters retain the zero extension off the units used in Section 10.

All parameters in the remainder of the section are fixed before \(x\) tends to infinity. Unless an independence is explicitly asserted, constants, thresholds, and rates of convergence may depend on those fixed parameters. The estimates are valid for arbitrary fixed marks, subject to the lower bounds on \(K\) in the Type II applications. We will choose the parameters together at the end.

Mass and distribution in progressions

Let \(r\) range over group integers, and define \[ \begin{split} J_0&=\sum_r\frac{\mathcal W(r)}r,\\ A_r&=\sum_{\substack{Q>x^{0.9}\text{ prime}\\crQ+1\equiv u\pmod M}} \Psi((crQ+1)/x),\qquad X_0=\sum_d w(d)=\sum_r\mathcal W(r)A_r. \end{split} \tag{188}\] We shall show that \(X_0\) is of order \(xJ_0/L\), and that the family at each small \(r\) has enough distribution to apply the block sieve. The harmonic sum makes this possible without constants depending on the pointwise bound \(B_K\).

Take temporarily \[ 0<\kappa<0.01,\qquad 0<\varepsilon<\kappa, \qquad 0<b<0.01, \qquad D=x^{1/2-\kappa/2}. \tag{191}\] For an odd squarefree integer \(\ell\), let \(A_r(\ell)\) denote the sum defining \(A_r\) with the additional condition \(\ell\mid crQ+1\), and set \[ g_r(\ell)=\frac{\mathbf{1}_{\{(\ell,Mr)=1\}}}{\varphi(\ell)},\qquad E_r(\ell)=A_r(\ell)-g_r(\ell)A_r. \tag{192}\] In particular, \(g_r(1)=1\) and \(E_r(1)=0\).

To express the sieve products, define \[ \begin{split} V_M(y)&=\prod_{\substack{2<p\le y\\p\nmid M}} \left(1-\frac1{p-1}\right),\\ \mathfrak S_M&=2\prod_{p>2}\left(1-\frac1{(p-1)^2}\right) \prod_{\substack{p\mid M\\p>2}} \left(1-\frac1{p-1}\right)^{-1}>0. \end{split} \tag{199}\] Division by \(V(y)\) and Mertens’ theorem give \[ V_M(y)\sim\mathfrak S_M V(y),\qquad V(y)\sim\frac{\mathrm{e}^{-\gamma_E}}{\log y}, \tag{200}\] where \(\gamma_E\) is Euler’s constant. The primes dividing any supported \(r\) are large enough that their omission changes these products negligibly. More precisely, (184) gives \[ \sum_{p\mid r}\frac1p\le0.11L^{0.9}\exp(-L^{0.1})=o(1), \qquad \prod_{2<p\le y}(1-g_r(p))=(1+o(1))V_M(y), \tag{201}\] uniformly for \(r\le x^{0.11}\) and all \(y\). The bound on the relative error is independent of \(K\).

Two sieve estimates

The block sieve of Lemma 32, from [20], now gives the initial mass without small prime factors and a conditional mass for a specified divisor. Both estimates use the distribution already proved, but at different sieve depths.

First sieve the prime objects \(Q\) in \(A_r\), for \(r\le x^\varepsilon\), by the conditions \(p\mid crQ+1\) at odd primes \(p\le z=x^b\). The local densities are \(g_r(p)\), with \(g_r(p)\le1/2\) and \(\sum_{v<p\le v^2}g_r(p)\ll1\) with absolute constants. Choose the even depth \[H=2\left\lfloor\frac1{40b}\right\rfloor.\] For sufficiently small fixed \(b\), this is large enough for the lower bound in Lemma 32; moreover, \[z^{4H+2}\le x^{0.2+2b}<D,\qquad \mathrm{e}^{-H}\le\mathrm{e}^2\exp(-1/(20b)).\] All remainders are therefore covered by (193). Every supported \(d\) is odd. Summing the sieve bound with the marks, using (200)–(201) and the negligible tail in Lemma 35, gives \[ S_b:=\sum_{P^-(d)>x^b}w(d) \ge\frac{\mathfrak S_M X_0}{L} \left\{\frac{\mathrm{e}^{-\gamma_E}}b (1-C_s\mathrm{e}^{-c_s/b})+o(1)\right\}, \tag{202}\] where \(C_s,c_s>0\) are absolute; the displayed bound on \(\mathrm{e}^{-H}\) permits \(c_s=1/20\), after enlarging \(C_s\). The remainders are \(o(X_0/L)\) by taking \(A>2\) in (193).

Removing composites by their least prime factor

We next pass from \(S_b\) to \[U_\kappa=\sum_{P^-(d)>x^{1/2-\kappa}}w(d).\] For a bin \((\gamma,\gamma']\subseteq(b,1/2-\kappa]\), with the first left endpoint allowed to equal \(b\), define \[T_{\gamma,\gamma'}= \sum_{\substack{x^\gamma<n\le x^{\gamma'}\\n\text{ prime}}} \sum_{P^-(m)>x^\gamma}w(mn).\] If \(x^\gamma<P^-(d)\le x^{\gamma'}\), taking \(n=P^-(d)\) and \(m=d/n\) shows that its weight is included in this sum. This remains true when \(n^2\mid d\).

Partition both factors into disjoint half-open dyads with endpoints powers of two, retaining pairs that meet the smooth product support and the range of \(n\). On that support, with \(t=\log n/L\), \[\frac{\log m}{L}=1-t+O(L^{-1}).\] Every retained full \(m\)-dyad therefore satisfies (186) with \[s_-=1/2+\kappa/2,\qquad s_+=1-b/2,\] for large \(x\); in particular \(s_->\gamma\). Both dyad scales are at least \(x^{b/2}\), and their product is comparable to \(x\). There are \(O(L)\) such pairs, since each \(n\)-dyad allows only a bounded number of \(m\)-dyads.

Apply Lemma 34 in every pair with \(\delta=b/2\) and with \(b_n\) the prime indicator restricted to its dyad and bin. These coefficients are \(W\)-rough. The total error is \(O(xL^{-5})=o(X_0/L)\), by (195) and \(J_0\ge1\). Hence \[T_{\gamma,\gamma'}= \sum_{\substack{x^\gamma<n\le x^{\gamma'}\\n\text{ prime}}} \sum_{m>x^\gamma}\frac{D_\gamma(\log m/L)}{LV(W)} \mathbf{1}_{\{P^-(m)>W\}}w(mn)+o(X_0/L).\] Joint continuity of the density on the fixed compact exponent range gives, on the support of the summand, \[D_\gamma(\log m/L) \le\sup_{\gamma\le t\le\gamma'}D_\gamma(1-t)+o(1).\] Since \(n>W\), the remaining roughness condition is precisely \(P^-(mn)>W\). Lemma 37 thus gives \[ T_{\gamma,\gamma'}\le\frac{\mathfrak S_M X_0}{L} \left\{\sup_{\gamma\le t\le\gamma'}D_\gamma(1-t) \log\frac{\gamma'}\gamma+o(1)\right\}. \tag{204}\]

The density estimate in Lemma 33 is normalized to match this expression: \[ \int_b^{1/2}D_t(1-t)\frac{dt}{t}=D_b(1)-1, \qquad D_b(1)\le\frac{\mathrm{e}^{-\gamma_E}}b(1+C_d\mathrm{e}^{-c_d/b}) \tag{205}\] for sufficiently small fixed \(b\). On \(b\le\gamma\le t\le1/2-\kappa\), the point \((\gamma,1-t)\) stays a fixed distance from \(s=\gamma\). Uniform continuity and the finite mass of \(dt/t\) therefore give a sufficiently fine fixed partition \[b=\gamma_0<\gamma_1<\cdots<\gamma_J=1/2-\kappa\] such that \[\begin{align*} \sum_{j=1}^J \sup_{\gamma_{j-1}\le t\le\gamma_j}D_{\gamma_{j-1}}(1-t) \log\frac{\gamma_j}{\gamma_{j-1}} &\le\int_b^{1/2-\kappa}D_t(1-t)\frac{dt}{t}+0.1\\ &\le D_b(1)-1+0.1. \end{align*}\] Subtracting (204) for these finitely many bins from (202) proves \[ U_\kappa\ge\frac{\mathfrak S_M X_0}{L} \left\{0.9-\frac{\mathrm{e}^{-\gamma_E}}b (C_s\mathrm{e}^{-c_s/b}+C_d\mathrm{e}^{-c_d/b})+o(1)\right\}. \tag{206}\] The unit term in (205) is what leaves a positive mass after this subtraction. It remains to remove the composites whose two prime factors are both close to \(\sqrt x\).

Composites with two nearly equal prime factors

A composite counted by \(U_\kappa\) has exactly two prime factors with multiplicity: three factors greater than \(x^{1/2-\kappa}\) would have product exceeding \(2x\). Each factor lies in \([x^{1/2-\kappa},2x^{1/2+\kappa}]\). Cover the ordered factor pairs by dyadic boxes \[ \mathcal B=[M_1,2M_1)\times[M_2,2M_2) \tag{207}\] that meet this range in each coordinate and admit a product in \([x,2x]\). They satisfy \(x/4\le M_1M_2\le2x\), and their full coordinate intervals lie in \([x^{0.46},x^{0.54}]\) for large \(x\). There are at most \(C_{\rm dyad}(\kappa L+1)\) boxes, where \(C_{\rm dyad}\) is absolute: this bounds the number of first coordinate dyads, and the product condition leaves only a bounded number of second coordinate dyads for each.

The rough-pair bound holds on a slightly wider fixed range of boxes, which includes all the boxes just selected.

Return now to the boxes selected in (207), whose full coordinate intervals lie in \([x^{0.46},x^{0.54}]\). We replace the prime indicators in each of these boxes. Apply Lemma 34 with \[\gamma=0.4,\qquad s_-=0.46,\qquad s_+=0.54, \qquad\delta=b/2.\] In this range \(R_{0.4}(m)\) is exactly the prime indicator, since \(2(0.4)>0.54\), and \(D_{0.4}(s)=1/s\). Its comparison coefficient is therefore \[B_{\rm pr}(m)=\frac{\mathbf{1}_{\{P^-(m)>W\}}}{V(W)\log m}.\] The identity \[\begin{aligned} \mathbf{1}_{\{m\text{ prime}\}}\mathbf{1}_{\{n\text{ prime}\}} -B_{\rm pr}(m)B_{\rm pr}(n) &=(\mathbf{1}_{\{m\text{ prime}\}}-B_{\rm pr}(m))\mathbf{1}_{\{n\text{ prime}\}}\\ &\quad+B_{\rm pr}(m)(\mathbf{1}_{\{n\text{ prime}\}}-B_{\rm pr}(n)) \end{aligned}\] allows two successive applications, interchanging the variables in the second. The companion coefficient is \(W\)-rough and bounded by one for large \(x\) in both cases. All coefficients retain their box restrictions. The total error over the boxes is \(O(xL^{-5})=o(X_0/L)\).

Since \(\log m,\log n\ge0.46L\), Lemma 38 bounds the resulting sum in one box by \((0.46)^{-2}C_1X_0/L^2\). The ordered prime-pair sum includes prime squares and may count other semiprimes twice, which is harmless for an upper bound. Summing the boxes gives \[ \sum_{\substack{P^-(d)>x^{1/2-\kappa}\\d\text{ composite}}}w(d) \le C_2(\kappa+L^{-1})\frac{X_0}{L}+o(X_0/L), \tag{213}\] where \(C_2=(0.46)^{-2}C_1C_{\rm dyad}\) depends at most on \(M,\Psi\), and is independent of \(K,\kappa,b,\varepsilon\).

Choice of parameters and completion

The selected cover and global representation

Denote the cubic Hilbert symbol at a place \(v\) by \((\,\cdot\,,\,\cdot\,)_{3,v}\). In the convention of Kazhdan and Patterson, the restriction of their determinant-modified cocycle to the diagonal torus is \[ \sigma_{c,v}\bigl(\operatorname{diag}(a,b), \operatorname{diag}(a',b')\bigr) =(a,b')_{3,v}(ab,a'b')_{3,v}^{c}, \qquad c\in\mathbb{Z}/3\mathbb{Z}. \tag{214}\] The root groups have their canonical splittings. These conventions are those of [15].

For the \(n\)-fold cover of \(\mathrm{GL}_d\), where \(d\) is the rank parameter, the center criterion of [15] requires \(z^{d-1+2dc}\in F_v^{\times n}\) for a scalar \(zI\). For \(d=2,n=3\) the exponent is \(1+4c\), which is zero modulo three exactly when \(c=2\). Moreover, (214) on two scalars is \[\sigma_{2,v}(zI,z'I)=(z,z')_{3,v}^{\,1+4\cdot2}=1.\] Thus their full preimage is central and the ordinary scalar section splits. Quotienting its image gives the asserted local covers of \(\mathrm{PGL}_2\). The adelic cover and its rational splitting are obtained by the reciprocity construction in [15]. The ordinary scalar section agrees with that splitting on rational scalars, so the quotient retains a rational splitting.

We check the datum in (27). A Hilbert symbol with a cube argument is one; the cocycle on scalar pairs is also one. Consequently the cube and scalar sections commute and multiply without any remaining cocycle. If an element has two presentations in (27), their two cube coordinates differ by a common scalar cube, so the displayed absolute-value ratio agrees. The formula is therefore a genuine character, trivial on the split scalar subgroup. On a coroot cube it gives \[\omega_v\bigl(s_v(\operatorname{diag}(t^3,t^{-3}))\bigr)=|t|_v.\] This is exactly the exceptional condition of [15]. The exceptional condition alone would only specify the square of the value on \(a(\varpi^3)\); the datum in the main text chooses the pure absolute-value value.

Use the idele norm in (27) to define the global character. A rational diagonal lying in the adelic cube torus times scalars has a rational ratio which is a cube at every completion. If that ratio were not a cube in \(F\), its nontrivial Kummer extension would have a nonsplit unramified prime by Chebotarev. Thus the ratio is a rational cube. The product formula now shows that our character is trivial on the intersection with the rational lift. This is the compatibility hypothesis in [15]. Extending by the trivial character on rational lifts gives the character on the rational maximal abelian subgroup of that lemma. As in [15], one chooses an extension on the product of local maximal abelian subgroups which agrees with the former character on their intersection. The character on the scalar center is unitary.

At finite places, the local quotient and subrepresentation assertions are [15]. At tame finite places, part (f) of that theorem supplies the spherical vector for the normalized unramified datum. For a tame place, the uniqueness criterion is \[d=n-1,\quad 2(c+1)=0\pmod n, \qquad\text{or}\qquad d=n\] by [15]. It applies to \((d,n,c)=(2,3,2)\). The residue construction and restricted tensor product are [15]. Since the tame local Whittaker spaces have dimension one, [15] gives a nonzero global Whittaker functional, and part (b) gives local uniqueness at every place. At the complex places this also follows directly from [15]. The inducing spaces, their exceptional images, and the automorphic residue all inherit the scalar action of \(\omega\). That action is trivial, so these representations descend to the scalar quotients.

On the quotient torus the section of \(a(y)\) has cocycle \((x,y)_{3,v}^{2}\). Since \(-1\) is a cube in \(F\), this is an alternating bimultiplicative law. At a tame place it is trivial on pairs of units, and its value on a uniformizer and a unit is the cubic residue symbol or its inverse, with a common orientation at all places. Choose the embedding \(\iota\) with the orientation giving (31). The canonical compact splitting agrees with the sections of unit diagonals, Weyl elements, and integral upper root elements by [15]. Conjugating by the Weyl element gives the same agreement for the lower root group. [15] gives relations for the coefficient functions defining local Whittaker functionals. [15] convert them to spherical Whittaker values with the modular factor and give the central step. The direct local calculation below proves (32) and the lift identities (30) in these conventions.

At a complex place the Hilbert symbol and the cover are trivial. [15] identifies the exceptional representation with the ordinary principal series when \(d\le n\). Our cube datum fixes the characters on all of \(\mathbb{C}^\times\), because every complex number has a cube root. Triviality on scalars removes the determinant character; equivalently any possible remaining finite-order continuous character of the connected group \(\mathbb{C}^\times\) is trivial. This gives the characters in part (iii) of 13. Their Bessel function is computed in the main text from the Jacquet integral. Finally, the two Bruhat terms in the constant term of the Eisenstein series are described by [15]. At the exceptional residue the surviving term belongs to the Weyl-conjugate induction, as in [15]. Its complex inducing character, including the modular factor, is \(|y|_v^{1/3}\); in particular it has no angular part.

The normalized global Whittaker expansion

Let \(\lambda\) be the nonzero global functional just obtained. The restricted tensor product is spanned by pure tensors, so there is a pure tensor \(\varphi^\circ\) with \(\lambda(\varphi^\circ)\ne0\). Its nondistinguished finite places form a finite set. The final \(S\) contains this set; the arithmetic data are constructed only after that final choice. For \(v\notin S\), varying only the \(v\)-component of \(\varphi^\circ\) gives a nonzero local Whittaker functional. Normalize it at \(\varphi_v^\circ\). If other finitely many components have already been varied, the resulting partial functional at \(v\) is still a Whittaker functional, and local uniqueness makes it a scalar multiple of the normalized one. Evaluating at \(\varphi_v^\circ\) identifies that scalar. Induction on the number of varied components proves (28). The same argument factors the complex components.

At a finite product of remaining places the functional descends to the tensor product of the local twisted Jacquet quotients. The twisted Jacquet quotient of the principal series is finite dimensional by [15]. Since \(\Theta_v\) is its quotient by Theorem I.2.9(b), its twisted Jacquet quotient is a quotient of that finite-dimensional space. A functional on the finite tensor product is therefore a finite sum of products. This is the finite-S assertion in 13.

This normalization accounts for the correction to the original global formula. The erratum withdraws an auxiliary spherical-functional vanishing assertion on pp. 119–120. It replaces stabilization of the auxiliary coefficient \(C_S\) by stabilization of \(C_ST(S)\), where in rank two \[T(S)=\prod_{\substack{v\in S\\v\nmid\infty}}(1+Q_v^{-1}), \qquad Q_v=\#(\mathcal O_v/\mathfrak p_v).\] The sums in Theorems II.2.2–II.2.3 become finite-\(S\) limits with the factor \(T(S)^{-1}\), and the same change applies on p. 130 [16]. The preceding argument begins with the nonzero automorphic functional and normalizes each exterior local functional at the distinguished spherical vector. It proves the product (28) without the withdrawn assertion or an unnormalized infinite product.

The rational splitting identifies every nonconstant Fourier coefficient with this Whittaker function. Conjugate \(n(x)\) by the rational lift of \(a(\nu)\) and change \(x\) to \(\nu x\); the adelic Jacobian is one. This gives (29). On a fixed compact set of \(g\), smoothness at infinity and invariance under a fixed finite open subgroup reduce the expansion along \(F\backslash\mathbb{A}_F\) to the smooth Fourier expansion on a compact real torus. Integration by parts in sufficiently many archimedean directions gives absolute local uniform convergence there.

Sections and the spherical induced vector

Fix an exterior finite place \(\mathfrak p\). In this appendix write \(\mathcal O=\mathcal O_{\mathfrak p}\), let \(\varpi\) be the chosen uniformizer, put \(Q=q_{\mathfrak p}\), and abbreviate \(\psi=\psi_{\mathfrak p}\), \(\chi=\chi_{\mathfrak p}\), and \(\mathcal C=\mathcal C_{\mathfrak p}\). Thus \(\psi\) has conductor \(\mathcal O\), additive measure gives \(\mathcal O\) volume one, and \(\chi\) is the sextic residue character on \(\mathcal O^\times\). In all finite sums it is extended by zero to the zero residue, including when its exponent is zero. The embedding of the cubic symbol has been chosen so that the effective torus cocycle is \[ \mathcal C(\varpi,u)=\chi(u)^{-2},\quad \mathcal C(u,\varpi)=\chi(u)^2,\quad \mathcal C(u,u')=1,\quad \mathcal C(\varpi,\varpi)=1 \qquad(u,u'\in\mathcal O^\times). \tag{215}\] The last two identities follow from tameness and alternation. In particular pure uniformizer powers multiply without a cocycle. All symbols with \(-1\) are one because \(-1\in F^{\times6}\).

We first record explicitly why the ordinary rank-one decompositions are valid for these lifts. See Kubota [18] for the local rank-one construction. In the convention of [15], the determinant-modified formula is \[\sigma_c(g_1,g_2)= \left(\frac{x(g_1g_2)}{x(g_1)}, \frac{x(g_1g_2)}{x(g_2)\det g_1}\right)_3 (\det g_1,\det g_2)_3^c, \quad x(g)= \begin{cases} g_{21},&g_{21}\ne0,\\ g_{22},&g_{21}=0. \end{cases}\] We take \(c=2\) and then quotient the split scalar subgroup. Substitution in this formula gives \[n(x)a(y)=a(y)n(x/y),\qquad wa(y)=a(y^{-1})w\] on the quotient. Before quotienting, the second equality has the extra scalar \(yI\). All its cocycles are symbols between powers of \(y\) and \(-1\), hence are one. For \(H\ne0\), the matrix identity \[wn(H)=n(H^{-1})\operatorname{diag}(-H^{-1},H)\bar n(H^{-1})\] also lifts exactly. The product of the first two factors has cocycle one, and multiplication by the last factor has cocycle \((H^{-1},-H)_3=1\). The middle diagonal is the product of the split scalar \(HI\) and \(a(-H^{-2})\), again without a cocycle. Thus \[ wn(H)=n(H^{-1})a(-H^{-2})\bar n(H^{-1}) \tag{216}\] on the quotient. If \(H^{-1}\in\mathcal O\), the last factor is in the compact splitting. The latter agrees with the sections on unit diagonals, \(w\), and integral upper root elements by [15]; conjugation by \(w\) gives agreement on the lower root group. The diagonal \(a(-1)\) can therefore be absorbed in a compact factor in any spherical calculation.

Let \(\sigma^-\) be the unramified representation of the covering torus with central character the Weyl conjugate of (27). Its norm exponent on \(a(y)\) is \(-1/6\). Modulo scalars, the subgroup of torus elements of valuation divisible by three is maximal abelian: the tame group \(F_{\mathfrak p}^\times/F_{\mathfrak p}^{\times3}\) has one valuation and one unit coordinate over \(\mathbb{Z}/3\mathbb{Z}\), and the commutator pairing pairs those coordinates nondegenerately. The torus construction in [15] realizes \(\sigma^-\) by induction from a character of this subgroup. Its index is three, so the unramified Heisenberg representation has dimension three. The canonical unramified extension is trivial on unit lifts [15]; the induced basis vector supported on that subgroup is therefore unit fixed. Call it \(e_0\), and set \[ e_j=Q^{-j/6}\sigma^-(a(\varpi^j))e_0,\qquad j\in\mathbb{Z}. \tag{217}\] The central element \(a(\varpi^3)\) acts by \(Q^{1/2}\), so \(e_{j+3}=e_j\). These three vectors form a basis. From (215) and unit invariance one obtains \[ \sigma^-(a(\varpi^j u))e_0 =Q^{j/6}\chi(u)^{2j}e_j \qquad(j\in\mathbb{Z},\ u\in\mathcal O^\times). \tag{218}\]

Use the vector-valued realization of normalized induction \[I^-=\operatorname{Ind}_{\widetilde B}^{\widetilde{\mathrm{PGL}}_2} (\sigma^-),\qquad f(n(x)a(y)g)=|y|_{\mathfrak p}^{1/2} \sigma^-(a(y))f(g).\] Its spherical vector \(f^\circ\) is determined by \(f^\circ(k)=e_0\) on the compact subgroup. The total norm exponent in this transformation law is \(1/2-1/6=1/3\). The compact-invariant line of this normalized unramified induction is one dimensional by [15]. By [15], the local exceptional representation is the irreducible subrepresentation of \(I^-\) and has a nonzero compact-invariant vector. That vector is a scalar multiple of \(f^\circ\), so the subrepresentation contains \(f^\circ\).

The shells of the Jacquet integral

For a smooth \(f\in I^-\), define the vector Jacquet functional by \[\mathscr J(f)=\lim_{L\to\infty} \int_{\varpi^{-L}\mathcal O}f(wn(x))\overline{\psi(x)}\,\,\mathrm{d}x.\] This limit stabilizes. Indeed, when \(|x|\) is sufficiently large, the right invariance of \(f\) makes \(\bar n(x^{-1})\) fix \(f\). The lifted Bruhat formula then expresses \(f(wn(x))\) as \[|x|_{\mathfrak p}^{-1} \sigma^-(a(-x^{-2}))f(1).\] Choose \(m\) so that \(1+\varpi^m\mathcal O\) consists of cubes and its unit lifts fix the vector \(f(1)\) in the smooth torus representation. On a shell \(x=\varpi^{-l}v\), the displayed expression is unchanged when \(v\) is replaced by \(vu\), \(u\in1+\varpi^m\mathcal O\): the norm is unchanged, all cocycles with \(u\) are trivial, and \(a(u^{-2})\) fixes \(f(1)\). Each such unit coset is an additive coset of \(\varpi^m\mathcal O\). For \(l>m\), the character \(\psi(\varpi^{-l}v)\) has zero integral on it. All sufficiently large shells therefore vanish. This argument applies to every smooth \(f\), including each of its right translates.

If \(t\in F_{\mathfrak p}\), choose a ball large enough to contain \(t\) and to realize the stable values for both \(f\) and \(\pi(n(t))f\). Translation \(x\mapsto x+t\) preserves that ball and gives \(\mathscr J(\pi(n(t))f)=\psi(t)\mathscr J(f)\). Thus each scalar coordinate of \(\mathscr J\) is a Whittaker functional. This is a direct stable version, at the present inducing parameter, of the Jacquet functionals introduced in [15].

For \(e\ge0\) put \(J_e=\mathscr J(\pi(a(\varpi^e))f^\circ)\); explicitly, \[ J_e=\int_{F_{\mathfrak p}} f^\circ(wn(x)a(\varpi^e))\overline{\psi(x)}\,\,\mathrm{d}x. \tag{219}\] The integral retains the stable interpretation just established. We now evaluate it shell by shell.

Put \(x=\varpi^eH\) and use the first two lift identities above. The measure contributes \(Q^{-e}\), the modular factor of \(a(\varpi^{-e})\) contributes \(Q^{e/2}\), and its torus norm factor contributes \(Q^{-e/6}\). Thus every shell has the common factor \[ Q^{-e+e/2-e/6}=Q^{-2e/3}. \tag{220}\] If \(H\in\mathcal O\), then \(wn(H)\) belongs to the compact subgroup, and \(\psi(\varpi^eH)=1\). This region contributes \(e_{-e}\) inside the common factor.

Suppose instead that \(H=\varpi^{-l}v\), with \(l\ge1\) and \(v\in\mathcal O^\times\). In (216) the last factor is compact, while the first acts trivially in the inducing transformation law. The modular factor of \(a(-H^{-2})\) is \(Q^{-l}\), and its torus norm factor is \(Q^{l/3}\). Furthermore, \[a(\varpi^{2l}v^{-2}) =\mathcal C(\varpi^{2l},v^{-2})^{-1} a(\varpi^{2l})a(v^{-2}),\qquad \mathcal C(\varpi^{2l},v^{-2})^{-1}=\chi(v)^{-8l}.\] The minus sign in \(-H^{-2}\) has no effect. Multiplication by \(a(\varpi^{-e})\) changes the basis index by \(-e\), without a cocycle between the pure uniformizer powers. Finally \(\,\mathrm{d}H=Q^l\,\,\mathrm{d}v\). It follows that this shell contributes, inside (220), \[ Q^{l/3}e_{-e+2l} \int_{\mathcal O^\times} \chi(v)^{-8l}\overline{\psi(\varpi^{e-l}v)}\,\,\mathrm{d}v. \tag{221}\] This accounts separately for the measure, modular, norm, and Hilbert-symbol factors.

Define the normalized prime Gauss sums \[g_j=Q^{-1/2}\sum_{x\bmod\mathfrak p} \chi(x)^j\psi(x/\varpi),\qquad j\in\mathbb{Z}.\] The powers in these sums retain the unit mask. Since \(\chi(-1)=1\), replacing \(\psi(x/\varpi)\) by its complex conjugate has no effect on \(g_j\). For \(j\not\equiv0\pmod6\), the usual finite-field Gauss identity gives \[|g_j|=1,\qquad g_jg_{-j}=\chi(-1)^j=1.\] In particular \(g_2g_4=1\). These identities follow, for example, by expanding the product of the two sums, substituting \(x=ty\), and using the additive sum in \(y\); the result is \(Q\chi(-1)^j\) before division by \(Q\).

Write \(I_{e,l}\) for the integral in (221). The character there is \(\chi^{-2l}\), since \(-8l\equiv-2l\pmod6\). Its exact values are \[ I_{e,l}= \begin{cases} 1-Q^{-1},&1\le l\le e,\quad 3\mid l,\\ 0,&1\le l\le e,\quad 3\nmid l,\\ Q^{-1/2}g_{-2l},&l=e+1,\quad 3\nmid l,\\ -Q^{-1},&l=e+1,\quad 3\mid l,\\ 0,&l>e+1. \end{cases} \tag{222}\] Indeed, for \(l\le e\) the additive character is trivial, so unit orthogonality gives the first two cases. For \(l=e+1\), integration over the residue classes gives \(Q^{-1}\) times the corresponding prime Gauss sum; the trivial character with its unit mask gives \(\sum_{x\ne0}\psi(x/\varpi)=-1\). For \(l>e+1\), split each unit class modulo \(\varpi^{l-e-1}\) into its \(Q\) lifts modulo \(\varpi^{l-e}\). The multiplicative character is unchanged on these lifts, whereas the additive factors sum to \(\sum_{t\bmod\mathfrak p}\psi(t/\varpi)=0\). This also proves the asserted stabilization of (219).

We can now sum the shells. For \(k\ge0\), the integral unit ball and the shells \(l=3,6,\ldots,3k\) have the same basis index and combined coefficient \[1+(1-Q^{-1})\sum_{j=1}^{k}Q^j=Q^k.\] For \(e=3k\), the terminal shell \(l=3k+1\) has coefficient \(Q^{k-1/6}g_4\) and basis vector \(e_2\). For \(e=3k+1\), the terminal shell \(l=3k+2\) has coefficient \(Q^{k+1/6}g_2\) and basis vector \(e_0\). For \(e=3k+2\), the terminal shell \(l=3k+3\) has coefficient \(-Q^k\) and the same basis vector \(e_1\) as the preceding combined contribution. Multiplying by (220) therefore gives \[\begin{align*} J_{3k} &=Q^{-k}(e_0+Q^{-1/6}g_4e_2),\\ J_{3k+1} &=Q^{-k}(Q^{-1/2}g_2e_0+Q^{-2/3}e_2) =Q^{-k-1/2}g_2(e_0+Q^{-1/6}g_4e_2),\\ J_{3k+2}&=0. \tag{223}\end{align*}\] The second equality uses \(g_2g_4=1\).

In particular \(J_0=e_0+Q^{-1/6}g_4e_2\ne0\). Choose a scalar coordinate nonzero on \(J_0\). Its Jacquet functional is nonzero on the exceptional subrepresentation, since that subrepresentation contains \(f^\circ\). The local Whittaker space is one dimensional by [15]. Hence every spherical Whittaker function normalized at one has the ratios in (223): \[ W(a(\varpi^{3k}))=Q^{-k},\qquad W(a(\varpi^{3k+1}))=Q^{-k-1/2}g_2,\qquad W(a(\varpi^{3k+2}))=0. \tag{224}\] The middle term is \(Q^{-k-1}\sum_x\chi(x)^2\psi(x/\varpi)\), as claimed. This computation agrees with the central-step statement of [15]: on \(a(\varpi^3)\) the Weyl-conjugate inducing character is \(Q^{1/2}\) and the modular half-character is \(Q^{-3/2}\), so their product is \(Q^{-1}\).

Negative valuations and the unit normalization

For \(e<0\), choose \(x\in\mathcal O\) for which \(\psi(\varpi^e x)\ne1\). The spherical vector is fixed by \(n(x)\), while the split root relation and Whittaker equivariance give \[W(a(\varpi^e)) =W(a(\varpi^e)n(x)) =\psi(\varpi^e x)W(a(\varpi^e)).\] Thus the value is zero. For a unit \(u\), the compact section of \(a(u)\) fixes the spherical vector. The torus law gives \[a(\varpi^e u)=\mathcal C(\varpi^e,u)^{-1}a(\varpi^e)a(u),\] where \(\mathcal C\) includes the scalar action on a genuine vector. Consequently \[ W(a(\varpi^e u))=\chi(u)^{2e}W(a(\varpi^e)). \tag{225}\] Together with (224), this proves (32).

There is a useful check that fixes the orientation without referring to a convention for the name of a cubic Gauss sum. At a tame place the residue character is unchanged when \(\varpi\) is replaced by \(\varpi u\). Substitution \(x=uv\) in the residue sum yields \[Q^{-1}\sum_{x\bmod\mathfrak p}\chi(x)^2\psi(x/(\varpi u)) =\chi(u)^2 Q^{-1}\sum_{v\bmod\mathfrak p} \chi(v)^2\psi(v/\varpi).\] This is exactly the \(e=1\) transformation in (225). Thus the Gauss phase and the unit factor are a single compatible normalization. Changing the sign of an additive Gauss argument is harmless because \(\chi(-1)=1\).