Proof. Fixing one of the finitely many classes at the bounded modulus above \(3\) reduces the claim to primitive group characters modulo a primary \(r\), with \(Nr\ll Q\). Let \(S(y)=\sum_n v_n e(\operatorname{Tr}(ny))\). Primitive Gauss-sum expansion and character orthogonality give \[\sum_{\chi\bmod r}^{\mathrm{primitive}} \left|\sum_n v_n\chi(n)\right|^2 \le \frac{\varphi(r)}{Nr} \sum_{\substack{h\bmod r\\(h,r)=1}}|S(h/r)|^2 \le \sum_{\substack{h\bmod r\\(h,r)=1}}|S(h/r)|^2.\] Fixed factors arising from the trace-dual lattice are immaterial. Distinct reduced fractions with denominator norms \(O(Q)\) are separated by \(\gg Q^{-1}\) on the dual torus: their difference, after a lattice translation, has a nonzero integral numerator and denominator of absolute value \(O(Q)\).

The planar additive large sieve for such a separated set has bound \(O(Z+Q^2)\). Here is the usual smoothing argument. Take a nonnegative Schwartz majorant of the disk containing the support of \((v_n)\), at scale \(\sqrt Z\). By duality and Poisson summation the resulting Gram matrix has entries bounded by \[C_N Z\sum_{\ell}(1+\sqrt Z\,|y-y'+\ell|)^{-N}.\] Packing a \(Q^{-1}\)-separated set in successive planar annuli bounds each row sum by \(O(Z+Q^2)\), for a fixed sufficiently large \(N\). The operator norm has the same bound, proving the assertion. The exponents \(1\) and \(2\) at the primes dividing \(pq\) distinguish the cubic characters in the last assertion; bounded ramified factors only introduce a bounded multiplicity. ◻

Proof. For the first estimate write uniquely \(n=s t^2 c^3\), with \(s,t\) coprime and squarefree. Restrict their norms to dyads \(S,V,C\), so that \(SV^2C^3\asymp Z\). For each fixed \(t,c\), the factors involving them have modulus at most one on the outer variable. Apply Theorem 3 to the \(s\) sum and then use the triangle inequality in the outer squared-sum norm. The squared norm of the restricted coefficient sequence is \(O_\epsilon(X^\epsilon S)\). There are \(O(VC)\) choices of \(t,c\), and the resulting bound is \[ X^\epsilon (VC)^2S \bigl(P+S+(PS)^{2/3}\bigr). \tag{11}\] Since \((VC)^2S\ll Z\) and \(S\ll Z\), summing the logarithmically many dyads proves (9). We retain (11) when a restriction makes \(S\) shorter.

For the second estimate factor the outer argument in the same way. Fixing \(t,c\) when \(S\ge V\), and applying Theorem 3 to \(s\), costs \[X^\epsilon VC\bigl(S+P+(SP)^{2/3}\bigr)\sum_b|u_b|^2.\] Fixing \(s,c\) when \(S<V\) gives instead \[X^\epsilon SC\bigl(V+P+(VP)^{2/3}\bigr)\sum_b|u_b|^2,\] with character conjugation if necessary. Using \(SV^2C^3\ll J\) and the indicated order of \(S,V\), each expression is bounded by the right side of (10). The coprimality restriction imposed by \(c\) is simply a restriction of the coefficients for each fixed \(c\). For a full-lattice argument write \(a=\zeta\lambda^i s t^2 c^3\), \(i\in\{0,1,2\}\), allowing powers of \(\lambda\) in \(c\). The finitely many unit and residual ramified factors are coefficient twists. The same proof applies. ◻

Derivation of the stated consequence. The summation formula and coefficient information are the cited background theorem, with Equations (5.7), (5.13)–(5.14), (5.74), and (5.79)–(5.81). In particular, the dual theta coefficient is read from the proof formula (5.79), where its index is \(\lambda^{-3}\nu\); this avoids the repeated ramified exponent in the printed (5.67). The theta formulas give \(|a_r(\nu)|\ll3^{\max(j,0)/6}|h'|\), which implies the stated weaker bound; the finitely many exceptional ramified cases have bounded factors. Multiplying the original summation formula by \(G(r)\) uses \(G(r)\overline{g(r)}=R^{1/2}\), where \(g(r)=\sqrt R\,G(r)\). This proves the displayed normalization of both terms.

We give the absolute convergence argument for (12). On \(\operatorname{Re}z=-\sigma\), rapid Mellin decay and Stirling’s formula bound the transform by \[X^{O(\sigma)}L^{O(1)} \bigl(N(d^3\nu)\bigr)^{-\sigma}.\] For a prime of norm \(p\) dividing \(r\), the exponents in \(h,h'\) are \(e\in\{0,1\}\) and \(k\ge0\). Its absolute local sum is bounded by \[\sum_{e=0}^1\sum_{k\ge0} p^{1_{e+k>0}}p^{k/2}p^{-(e+3k)(1+\sigma)} =1+p^{-\sigma}+O_\sigma(p^{-3/2-3\sigma}).\] For every fixed \(\eta>0\), the product over \(p\mid r\) is \(O_{\sigma,\eta}(R^\eta)\): separate the finitely many small primes, then bound each remaining factor by \(p^\eta\). The free \(w\) sum converges at \(1+\sigma\), the ramified sum is dominated by \(\sum_{j\ge0}3^{-j(2/3+\sigma)}\), and the \(d\) sum converges at \(5/2+3\sigma\). Choosing \(\sigma\) and \(\eta\) sufficiently small in terms of the final \(\epsilon\) proves (12). ◻

Proof. Nonvanishing of \(\mathcal F_z(n)\) requires every prime factor to have norm greater than \(z\). Three factors would give \(N(n)>X^{1.20}\), outside the fixed envelope. On a prime in that envelope the factor \(\mathcal F_z\) is exactly \(1\). This proves (49).

Expand, independently at every prime factor, \[1-\psi_z=(1-\psi_w)+(\psi_w-\psi_z).\] The chosen factors in the second summand form an unordered subset; ordering them and dividing by \(i!\) gives (50). The difference of cutoffs vanishes below \(w\) and above \(2z\). Since the total norm is \(O(X)\) and every chosen prime has norm at least \(X^\xi\), their number is bounded in terms of \(\xi\). The same bound applies to the number of prime factors of \(h\) when \(\mathcal F_w(h)\ne0\). ◻

Proof. Normalize the logarithms of the prime scales by the logarithm of their product. In any limiting configuration they are positive, sum to \(1\), and are bounded below in terms of \(\xi\). On ordinary blocks no part reaches \(2/3\), since every part is at most \(1-.345=.655\).

We claim that a positive partition of \(1\), with every part less than \(2/3\), has a subset sum in \((1/3,2/3)\) unless it is \((1/3,1/3,1/3)\). If there is no such subset, every individual part is at most \(1/3\). If a part \(a<1/3\) exists, start with it and add parts until the sum first exceeds \(1/3\). The resulting sum is at most \(2/3\). Equality would require both the preceding sum and the last added part to be \(1/3\); but then \(a\) together with that last part is a subset sum in \((1/3,2/3)\). Thus equality is also impossible. The only remaining partition has all parts equal to \(1/3\).

There are finitely many possible factor counts. On the compact sets of their allowed exponent vectors, outside a fixed small neighborhood of the exceptional three-part vector, an interior subset has a uniform distance from both endpoints. Group that subset and its complement and call the shorter group \(B\). The fixed support ratios can be absorbed by reducing the uniform gap, giving the claimed \(\delta_1\). In the exceptional neighborhood, take the largest prime scale for \(B\); make the neighborhood small enough that \(B<X^{.40}\) and \(B\ll\sqrt X\).

For upper blocks, a part exceeding \(2/3\) can only come from \(h\), and its complement has limiting exponent at least \(s_0\). If no part exceeds \(2/3\), a subset sum in \([1/3,2/3]\) exists by the same successive-sum argument. Fixed support ratios and limiting errors are absorbed by the spare \(\kappa\) in the asserted lower bound. ◻

Proof. Fix the last bin, its lower endpoint \(\ell\), and the numbers \(k_+\ge1\) and \(k_-\ge0\) of selected and remaining factors in this bin. All factors in earlier bins belong to \(m_+\) and all factors in later bins to \(m_-\). If \(Z\) is the applicable threshold, the crossing predicate is \[ N(r)S(m_+)/\ell<Z\le N(r)S(m_+). \tag{53}\] For an ordinary stop below the roughness boundary, additionally require \(N(r)S(m_+^{>})<X^{.36}\), where \(m_+^{>}\) contains the factors above that boundary. Every factor in those earlier bins is already on the selected side. Hence both predicates depend only on the \(b\)-side variables.

The surrogate uses the same \(\ell\) for every prime in the final bin. For a fixed original squarefree \(m\), exactly \(\binom{k_++k_-}{k_+}\) selections have the stated counts, and either all or none satisfy the crossing test. Give each selection the scalar \[ \binom{k_++k_-}{k_+}^{-1}. \tag{54}\] Their sum restores the original coefficient exactly. The Mobius factor and all prime weights split multiplicatively between \(m_+\) and \(m_-\). Once the last bin and its two counts have been fixed, the \(b\) coefficient sums only over \(r,m_+\), and the \(a\) coefficient only over \(m_-,u\).

More explicitly, let \(v(r)\) denote the weighted distinguished tuple sum, including its factor \(1/i!\), and put \(q(d)=\mu(d)\prod_{\pi\mid d}\psi_w(\pi)\). If \(j\) is the last bin and \(\mathcal C_{j,k}(r,d)\) is the crossing condition with exactly \(k\) factors of \(d\) in bin \(j\) and no factors in later bins, define \[\beta_{j,k}(b)=\sum_{rd=b}v(r)q(d) \boldsymbol1_{\mathcal C_{j,k}(r,d)}, \qquad \alpha_{j,l}(a)=\sum_{eu=a}q(e) \boldsymbol1_{\substack{e\text{ has factors only in bins }j,j+1,\ldots\\ e\text{ has exactly }l\text{ factors in bin }j}}.\] Then the stopped sum against any squarefree-supported kernel \(K\) is exactly \[\sum_{j,k\ge1,l\ge0}\binom{k+l}{k}^{-1} \sum_{a,b}\alpha_{j,l}(a)\beta_{j,k}(b)K(ab).\] Here \(K\) includes the product envelope, and the notation \(\mathcal C_{j,k}\) includes the failed first-stage test when needed.

One may enlarge these independent sums by allowing overlaps among \(r,m_+,m_-,u\). Each added tuple has nonsquarefree total product and is individually annihilated by both kernels, before coefficient collection. After collection, \(a,b\) can be restricted separately squarefree; a remaining cross-side common prime is killed by \(G(ab)\) and by \(\mu^2(ab)\). Thus independence does not require an unaccounted coprimality restriction.

The bin groups are unordered divisors. Only the already bounded distinguished tuple is ordered. Their multiplicities are therefore bounded by fixed powers of \(\tau(a),\tau(b)\), and the scalar (54) is at most \(1\). There are \(O(L)\) choices for the final bin and \(O(L)\) for each of its two counts. The complete bin occupancy vector is not an outer index: all other bin conditions remain inside the respective coefficient. Adding the boundary \(L^{C_r}\) adds at most one bin and changes none of these powers. ◻

Proof. First discard terms all of whose prime factors have norm at most \(y=\exp(\sqrt L)\). Apply Rankin’s inequality with \(\sigma=1/\sqrt L\). Fixed divisor multiplicities can be included in the Euler product; its logarithm is \(O(\log L)\), since \[\sum_{N(\pi)\le y}N(\pi)^{-1+1/\sqrt L}=O(\log L).\] As \(B\) is a fixed positive power of \(X\), the resulting total absolute weight is at most \(B\exp(-c\sqrt L)L^C\). This gives every desired logarithmic saving.

In the remaining terms select a largest prime, breaking ties by a fixed order. Fix its role, its norm dyad \(P\), and all other factors. There are boundedly many distinguished roles and one unordered selected-bin role; no permutation of a bin product is introduced. We have \(P\gg\exp(\sqrt L)\).

Refine its range by the bins. If the variable prime lies in \(m_+\), all surrogate data are constant within its bin. If it lies in \(r\), then \(N(r)\) is a fixed multiple of its norm. Accordingly, (53), the possible failed first-stage test, the final \(b\) dyad, and the largest-prime condition each give an interval restriction in its norm. Their intersection has only polynomially many interval pieces as the bin varies. After the other factors are fixed, the Mobius sign is constant, and the remaining prime weights are smooth with bounded scaled derivatives.

The prime ideal Siegel–Walfisz estimate from the background inputs applies on this prime range. A conductor bound \(L^C\) is a bound \((\log P)^{2C+O(1)}\), and arbitrary powers of \(\log P\) give arbitrary powers of \(L\) in the saving. Partial summation absorbs the smooth weights and any fixed-polylogarithmic norm twist. It applies to arbitrary subintervals by subtracting two initial-interval estimates.

Tie decisions and distinctness cause no loss: in this quadratic field there are at most two prime ideals of any prime norm, and at most one of an inert prime-square norm. Once the other factors are fixed, tie rules affect only endpoint norms. Coprimality to a number of polynomial norm removes \(O(L)\) additional primes. These deletions are negligible relative to \(P L^{-D}\), since \(P\gg\exp(\sqrt L)\).

For a fixed dyad \(P\), the total absolute weight of complementary factorizations is at most \((B/P)L^{O(1)}\), by the fixed divisor multiplicity bounds. Summing the prime cancellation over these choices, over roles, and over the polynomially many dyads and interval pieces proves the assertion. The argument for pure prime-tuple coefficients is the same, without the surrogate restrictions. ◻

Completion of Proposition 15. The identities (49), (50), and (51) are exact on squarefree numbers. Lemma 17 treats the large-\(R\) branch, and Lemma 18 treats the small-\(R\) branch. The actual norm ranges proved above give parts (1) and (2). Lemma 19 gives part (3), and the three-prime counting argument gives part (4). Ordinary stopped terms have \(H\gg X^{.08}\), so no additional small-\(H\) configuration is omitted. The sparse estimate gives part (5).

The total number of pieces is a fixed power of \(L\): the distinguished tuple has bounded length, the bin and endpoint-count indices have polynomially many choices, and the side dyads add only polynomial factors. No full occupancy vector of the bins is enumerated. The exponents in this complexity and in the divisor bounds are independent of \(C_r,C_g\); those parameters affect only fixed constants in the exceptional triple count, not its polynomial dependence on \(\log L\).

The parameter choices are therefore noncircular. First fix the absolute power-width constant for \(B\le X^{.40}\) in Proposition 13; then fix \(\kappa,\rho,\xi\) and the resulting bounded tuple complexity. Choose the logarithmic output and localization savings, and the large-height threshold power, after that complexity. Finally choose \(C_g,C_r\) and the needed SW saving powers. The large-\(B\) prime-polynomial estimates may depend on the now fixed tuple complexity and \(\xi\); they do not determine the earlier absolute power-width constant. ◻

Proof. First omit the product envelope. In the notation of Proposition 13, multiplicativity gives \[\sum_{a,b}\alpha_a\beta_bG(ab)N(ab)^{it} =\sum_a\alpha_aG(a)N(a)^{it}F_t(a).\] Apply Cauchy’s inequality in \(a\) to the corrected estimate (31). Since \(\sum_a|\alpha_a|^2\ll AL^{O(1)}\), its contribution is \[\ll (A\mathcal Q)^{1/2}L^{-K/2+O(1)} \ll X^{5/6}L^{-K/2+O(1)}.\] The resulting main term is \[c_*\sum_a\alpha_a\mu^2(a)N(a)^{it-1/6} \sum_b\beta_bN(b)^{it-1/6}.\] It remains to reinstate coprimality between \(a\) and \(b\). Any common prime has norm greater than \(L^{C_r}\). The number of pairs sharing such a prime is \[\ll AB\sum_{N\mathfrak p>L^{C_r}}\frac1{N\mathfrak p^2} \ll AB L^{-C_r}.\] Here ideals of norm at most \(Y\) have count \(O(Y)\), so the last inequality follows even on summing over all ideals. By Cauchy’s inequality and the coefficient moment bounds, the weighted number of these pairs is \(\ll AB L^{-C_r/2+O(1)}\). Multiplication by \(N(ab)^{-1/6}\) makes their model contribution \(\ll X^{5/6}L^{-C_r/2+O(1)}\). Taking \(K,C_r\) large proves (56) without its envelope.

For the transition, the specified prime dyads and their bounded weights give \[\sum_a|\alpha_a|^2\ll\frac A{L^2},\qquad \sum_b|\beta_b|^2\ll\frac BL,\qquad \sum_b|\beta_b|\ll\frac BL.\] The possible ordering multiplicity for the two primes in \(a\) is bounded. Apply (32) and Cauchy’s inequality. The diagonal contributes \[\left(\frac A{L^2}\frac{AB}L\right)^{1/2} \ll X^{5/6}H^{-1/6}L^{-3/2},\] and the remaining displayed main bound contributes \[\left(\frac A{L^2}\frac{\mathcal Q}{L^2}\right)^{1/2} \ll X^{5/6}L^{-2}.\] The \(L^{-K}\) remainder is smaller by increasing \(K\). Since \(H\gg1\), this bounds the Gauss-sum term as asserted. The model term is bounded directly by \(O(X^{5/6}L^{-3})\), using the independent counts on the three specified prime dyads.

For completeness, the envelope introduces no logarithmic loss in these arguments. Mellin inversion separates \(V(N(ab)/X)\) into common norm twists with a rapidly decreasing, integrable weight. Keep shifts \(|v|\le L^E\), for a sufficiently large fixed \(E\), in the low-height estimates, choosing their permitted height exponent after \(E\). From \(L^E\) up to \(X^c\), with a small fixed \(c>0\), the coarse part of Proposition 13 and Cauchy’s inequality bound the uncorrected sum by \(X^{5/6}L^{O(1)}\); the model has the same bound trivially. Rapid decay of the Mellin weight makes this tail as small a log power as needed. Beyond \(X^c\), its rapid decay and the trivial bound \(XL^{O(1)}\) give a power saving. This proves both claims with the envelope. ◻

Proof. When \(B>X^{.40}\), apply Cauchy’s inequality in \(a,t\), followed by (33). The diagonal costs \[\ll A\sqrt B\,L^{O(1)} \asymp \frac X{\sqrt B}L^{O(1)} \ll X^{.80}L^{O(1)},\] whereas the error costs \(X^{5/6}L^{-K/2+O(1)}\). The product envelope separates by Mellin inversion. Its shifts may be truncated at a small fixed power of \(X\), since the tails are negligible trivially. The retained shifts satisfy \(2T+|u_0|\le X^{.17}\), because \(\rho<.001\). This proves the assertion in this range.

Assume now \(B\le X^{.40}\). Partition the two fixed log-norm intervals into disjoint cells of width \(1/J\), where \[J=L^{D'}\quad\text{in case (a)},\qquad J=X^\gamma\quad\text{in case (b)}.\] There are \(O(J)\) cells on each side. Put \[a_i=\|\alpha\,1_{\text{cell }i}\|_2, \qquad b_j=\|\beta\,1_{\text{cell }j}\|_2.\] A smooth majorant of an \(a\)-cell has derivatives of order \(s\) bounded by \(O_s(J^s)\). Its lattice-point count is \[O(A/J+\sqrt A+1)=O(A/J),\] because \(A\gg X^{.60}\) and \(\gamma<.30\). Restricting \(\beta\) sharply to a cell preserves its divisor bounds and squarefree support; there is no additional prime-polynomial hypothesis in this range of \(B\).

For a fixed cell pair, Cauchy’s inequality and the averaged dispersion estimate give the bound \[ C a_i\left(\frac A J b_j^2+\mathcal Q\varepsilon_J\right)^{1/2}, \qquad \varepsilon_J= \begin{cases} L^{-K},&\text{in case (a)},\\ X^{-10\gamma},&\text{in case (b)}. \end{cases} \tag{60}\] It holds uniformly after any fixed common Mellin shift: for \(B\le X^{.40}\), Proposition 13 places no restriction on \(u_0\). In particular it applies with the product envelope, whose Mellin transform has bounded \(L^1\) norm.

Call a pair near if its rectangle meets \(\left|\log(N(ab)/X_0)\right|\le c_0/J\), with a fixed \(c_0>0\). Every cell has only \(O(1)\) near neighbors. Hence the diagonal parts of (60), summed over near pairs, are \[ \ll\sqrt{\frac A J}\,\|\alpha\|_2\|\beta\|_2 \ll X^{5/6}L^{O(1)}H^{-1/6}J^{-1/2}. \tag{61}\] This is the required log saving in case (a), by choosing \(D'\) large. In case (b), \(H\gg X^{-9\kappa}\), so its additional power of \(X\) is at most \(3\kappa/2-\gamma/2<0\).

The errors in (60) are absolute errors; no reduction with the mass of a \(b\)-cell is assumed. Even summing them over all \(O(J^2)\) pairs gives at most \[\begin{align*} \sqrt{\mathcal Q\varepsilon_J}\,J\sum_i a_i &\ll \sqrt{\mathcal Q\varepsilon_J}\,J^{3/2}\|\alpha\|_2 \\ &\ll X^{5/6}L^{O(1)}J^{3/2}\varepsilon_J^{1/2}. \tag{62}\end{align*}\] In case (a), take \(K\) large after \(D'\). In case (b), the additional factor is \(X^{-7\gamma/2}\).

It remains to handle far pairs. Set \(\ell=\log(N(ab)/X_0)\). On a far pair \(|\ell|>c_0/J\). Integrating by parts \(q\) times in the height integral differentiates only \(h_T\) and produces a factor \((J/T)^q\), a new height function proportional to \(T^q h_T^{(q)}\), and the product weight \((J\ell)^{-q}\). The new height function still has the same normalized derivative bounds.

There is no loss by a power of \(J\) in the Mellin \(L^1\) norm. Indeed, for a fixed integer \(q\ge2\), choose a smooth function \(F_q\) on \(\mathbb R\) which equals \(s^{-q}\) for \(|s|\ge c_0\) and is zero near zero. Both \(F_q\) and \(F_q''\) are integrable. Consequently its Fourier transform is integrable, and \[\|\widehat{F_q(J\,\cdot)}\|_1=\|\widehat F_q\|_1<\infty.\] The smooth extension \(F_q(J\ell)\), multiplied by the fixed envelope, therefore has Mellin \(L^1\) cost \(O_q(1)\), uniformly in \(J\). All these shifts are independent of \(t\). Even a shift larger than \(T\) is allowed here: it simply multiplies the fixed Dirichlet-polynomial coefficients by phases, as reflected in the unrestricted \(u_0\) assertion of Proposition 13.

Apply (60) again with this separated weight. Before the integration-by-parts factor, the sum of all diagonal parts is \[ \ll\sqrt{\frac A J}\left(\sum_i a_i\right) \left(\sum_j b_j\right) \ll X^{5/6}L^{O(1)}H^{-1/6}J^{1/2}. \tag{63}\] In case (a), \((J/T)^q\le L^{-q(C_t-D')}\), which pays for this loss by taking \(C_t\) sufficiently large. In case (b), it is at most a constant times \(X^{-q(.01-\gamma)}\), which pays for \(H^{-1/6}J^{1/2}\ll X^{3\kappa/2+\gamma/2}\). The errors are already controlled by (62). This proves (59). ◻

Proof. With the Fourier convention in (64), the Fourier transform of \(1_{[0,\log2]}\) is \((1-2^{-it})/(it)\), interpreted continuously at zero. The inverse transform of \(\eta(t/T)\) is a Schwartz approximate identity \(T k(Ts)\) of integral one. Convolving the interval indicator with it proves (64). Away from either endpoint, the error is bounded by the integral of the rapidly decreasing kernel beyond that endpoint; this proves (65). In the portion where \(V\ne1\), the distance to \([1,2]\) is bounded below, so the same estimate holds there. On \(1/2\le v\le4\), distance in \(\log v\) is comparable to distance in \(v\). Summing \((1+T|n-X_0|/X)^{-2}\) over integers, for \(X_0=X,2X\), costs \(O(1+X/T)\), uniformly in the location of \(X_0\). This gives (66), including exact endpoint equality. ◻

Proof of Theorem 1. Use Lemma 22 with \[T_{\max}=X^{1/6+\rho}.\] At each norm the number of primary prime generators is divisor-bounded, and \(|G(\pi)|=1\). Thus the lemma applies separately to the Gauss-sum and model prime coefficients. Taking \(\epsilon<\rho\) in (66), their total truncation error is \[\ll X^\epsilon(1+X/T_{\max}) =o\!\left(\frac{X^{5/6}}L\right).\] It remains to estimate their difference with the smoothed indicator.

Split the integral in (64) into bounded frequencies and smooth dyadic height blocks. Outside bounded frequencies its two endpoint terms have the form \[\int h_T(t)\left(\frac{N(n)}{X_0}\right)^{it}\frac{dt}T, \qquad X_0=X\text{ or }2X,\] where \(h_T^{(j)}\ll_jT^{-j}\). Indeed the factor \(1/(it)\) is \(1/T\) times a smooth bounded function on a height block; the cutoff at \(T_{\max}\) preserves these derivative bounds. At bounded frequencies the original difference of endpoint terms is bounded, including at zero.

For \(|t|\le L^{C_t}\), use the ordinary-block decomposition of Proposition 15 on the difference \(\mathcal D(n)\). The Type I terms are controlled by Proposition 8; a norm twist of logarithmic height only introduces logarithmic derivative bounds. The regular Type II terms are controlled by Proposition 20 with the arbitrary output power \(D\) fixed above. The sparse terms have a fixed power saving.

The remaining terms are exactly the three-prime transition described in Proposition 15. They have bounded weights, \(1\ll H<L^{C_g}\), and only \(O((1+\log L)^{C})\) configurations for a fixed \(C\). Proposition 20 bounds each by \(X^{5/6}L^{-3/2}\). There are only \(O(1+\log L)\) low-height blocks. Therefore their aggregate contribution is \[\ll X^{5/6}L^{-3/2}(1+\log L)^{C+1} =o\!\left(\frac{X^{5/6}}L\right).\] All other low-height terms are smaller after summing their \(L^{C_{\mathrm{dec}}}\) pieces.

At heights \(T\ge L^{C_t}\), first discard the undecomposed prime model. On the fixed envelope, write its norm polynomial as \[P_0(t)=\sum_{n\asymp X}d_n n^{it},\qquad \sum_n|d_n|^2\ll X^{2/3}L^{O(1)}.\] The last bound follows by collecting the model coefficients \(c_*N(\pi)^{-1/6}V(N(\pi)/X)\) at equal norms and using the divisor moment bound. The ordinary mean-value inequality and Cauchy’s inequality give \[\begin{align*} \int_{T\le|t|\le2T}|P_0(t)|\frac{dt}T &\ll X^{1/3}L^{O(1)}(1+X/T)^{1/2}\\ &\ll X^{5/6}L^{O(1)}T^{-1/2}, \end{align*}\] since \(T\ll X^{1/6+\rho}<X\). Taking \(C_t\) large makes this negligible even after all \(O(L)\) height blocks. No decomposition of this model term, and no cancellation between its decomposed pieces, is required.

Apply the decomposition to the Gauss-sum term alone at the remaining heights. Ordinary blocks have \(T\ll X^{.01}\); their no-stop Type I terms satisfy \(R\le X^{.40}\). By Proposition 9, the first term of their bound has exponent at most \[\frac12+\frac34(.40)+\frac12(.01)=.805<\frac56.\] For upper blocks \(T\gg X^{.01}\), use the upper-block decomposition. Its no-stop terms have \(R\le X^{1/3-\kappa/2}\), and their first-term exponent is \[\frac12+\frac34\left(\frac13-\frac\kappa2\right) +\frac12\left(\frac16+\rho\right) =\frac56-\frac{3\kappa}{8}+\frac\rho2<\frac56.\] The pole term in Proposition 9 is \(X^{5/6}L^{O(1)}T^{-q}\) for every fixed \(q\), and is therefore negligible above \(L^{C_t}\). These bounds permit the uniformly smooth envelope at scale \(X/N(r)\), and any twist of the outer coefficient has absolute value one.

Every nonsparse Type II term is in case (a) or case (b) of Proposition 21. Its bound can have the arbitrary fixed output power \(D\). The sparse ordinary-block terms retain their power saving uniformly in the norm twist. Summing the \(O(L)\) height blocks and the at most \(L^{C_{\mathrm{dec}}}\) pieces consequently costs less than the chosen logarithmic saving. Changing the decomposition between height blocks is legitimate because each is an exact identity on squarefree arguments.

Combining the low and high heights with the truncation error proves \[ \sum_{X<N(\pi)\le2X} \left(G(\pi)-c_*N(\pi)^{-1/6}\right) =o\!\left(\frac{X^{5/6}}{\log X}\right). \tag{67}\] Changing endpoint inclusion has only divisor-bounded cost. Dyadically sum (67). For example, on dyads above \(X^{1/2}\), the little-oh coefficient is uniformly small, the logarithms are comparable to \(\log X\), and the \(5/6\)-powers form a geometric sum. The terms below \(X^{1/2}\) are \(O(X^{1/2})\) trivially. We obtain \[\sum_{N(\pi)\le X}G(\pi) =c_*\sum_{N(\pi)\le X}N(\pi)^{-1/6} +o\!\left(\frac{X^{5/6}}{\log X}\right).\] Finally the prime ideal theorem in the fixed field gives \(\#\{\pi:N(\pi)\le y\}\sim y/\log y\); omission of the prime above \(3\) is immaterial. Partial summation yields \[\sum_{N(\pi)\le X}N(\pi)^{-1/6} \sim\int_2^X\frac{u^{-1/6}}{\log u}\,du \sim\frac65\frac{X^{5/6}}{\log X}.\] This proves the asserted primary-prime asymptotic with constant \(\frac65c_*\). ◻

Proof. Rajan [14] gives the unitary completion, contragredient functional equation and entire-order-one assertion. There is one complex place, field discriminant \(3\), angular parameter \(\ell\) and no imaginary norm parameter. Its gamma factor and conductor factor give exactly (70). The dual angular order is \(-\ell\). A nonzero angular component cannot be removed by multiplying by a finite character or an imaginary norm power: both are trivial on the connected archimedean unit circle, whereas its \(\ell\)th angular character is nontrivial for \(\ell\ne0\). Hence \(\Psi\) is not principal or norm-equivalent to a real quadratic character when \(\ell\ne0\). For \(\ell=0\), a nonprincipal cubic character is nonreal. There is therefore no exceptional real zero in either case.

For clarity we give the uniformity deduction for (71). Apply the published [9], which explicitly restates the prime-number estimate of [8]. Viewed over rational primes, \(L(z,\Psi)\) has degree \(m=2\) and local parameters of modulus at most one. In particular its logarithmic-derivative coefficients satisfy \(|a(p^j)|\le2\), and the required second-moment hypothesis follows from \(\sum_{n\le x}|a(n)|^2\Lambda(n)^2\le4x(\log x)^2\). Gamma duplication gives conductor \(q=3D\), parameters \(\mu_1=k\), \(\mu_2=k+1\), and analytic conductor \(\mathfrak C=3D(k+3)(k+4)\). Absolute convergence is supplied by the Euler product, and the functional-equation hypothesis by (70). The zero-free-region hypothesis is Rajan’s Proposition 1; the same conductor and infinity-parameter dependence is recorded in [1]. The absence of an exceptional character just proved permits the bound \[x\bigl(\log(x\mathfrak C)\bigr)^4 \exp\!\left(-\frac{c\log x} {\log\mathfrak C+\sqrt{\log x}}\right) +O(\sqrt x).\] This is \(O_{\ell,A,M}(x\log^{-M}x)\) for every \(M\) in the stated conductor range. Removing higher prime-ideal powers from the logarithmic derivative costs \(O(\sqrt x\log^2x)\) and proves (71). ◻

Proof. If \(V\le\sqrt R\,T^2\), collect equal rational norms of \(ud^3\). Divisor multiplicities bound the squared coefficient sum by \[X^\epsilon\sum_d Nd\, \#\{u:N(ud^3)\asymp V\} \ll X^\epsilon V\sum_d(Nd)^{-2} \ll X^\epsilon V.\] The ordinary mean-value inequality bounds the mean absolute size of the full sum by \(X^\epsilon\sqrt V\sqrt{1+V/T}\), which is at most the right side of (74). The possible main term is \(O_D(V^{5/6}R^{-1/6}T^{-D})\). Its ratio to the claimed bound is at most \(R^{-1/4}T^{1/6-D}\le1\) for \(D\ge1\), by the current inequality \(V\le\sqrt R\,T^2\). Thus the pole-subtracted claim also holds in this branch.

Suppose now \(V>\sqrt R\,T^2\). In the angular transform use \(W(x)x^{it}\) and the outside scalar \(V^{it}\); its Mellin factor is \(\mathcal MW(z+it)\). On \(\mathop{\mathrm{Re}}z=-A\), the gamma quotient grows as \((1+|\Im z|)^{2+4A}\) for fixed \(\ell,A\). For any sufficiently small fixed \(\delta>0\), truncate at \[N(d^3\nu)\le I_{\max}=X^\delta R^2T^4/V.\] This cutoff uses the dyadic height \(T\) and is independent of \(t\). The omitted tail is smaller than any prescribed negative power of \(X\) by taking \(A\) sufficiently large. Indeed the transform contributes the power \((R^2T^4/(VN(d^3\nu)))^A\); the extra \(T^2\) and the absolutely summed coefficients are absorbed by \(X^{-\delta A}\) after all polynomial scale bounds are fixed. Rapid decay in the translated Mellin variable controls its remaining tail. Partition the retained finite sum into norm dyads \(I\), so \[I\ll X^\delta R^{3/2}T^2.\] More explicitly, put \(D_0=R^2T^4/V\). The absolutely summed coefficients with the additional weight \(N(d^3\nu)^{-\sigma}\) cost \(O_{\sigma,\eta}(R^\eta)\). For \(A>\sigma\) the discarded part is therefore \(O_{\ell,A,\sigma,\eta}(R^{1/2+\eta}T^2D_0^\sigma X^{-\delta(A-\sigma)})\). All factors preceding the negative power have fixed polynomial bounds, justifying the stated choice of \(A\).

Move each finite transform to \(\mathop{\mathrm{Re}}z=1/2\). Its first numerator pole has real part \(5/6+|\ell|/2>1/2\), and \[|Q_\ell(1/2+iy)|=1.\] Translating the imaginary variable makes the weight \(\mathcal MW(1/2+iv)\) and the norm exponent \(i(v-t)\). The gamma quotient and conductor phase are scalars for the whole frequency polynomial. Minkowski’s inequality thus uses a common rapidly decreasing \(L^1\) majorant independent of \(t\). On this line the absolute coefficient, before summing frequencies, is a fixed constant times \[\sqrt V\,R^{-1/2} \frac{|a_r(\nu)b_r(\nu)|}{Nd\sqrt{N\nu}}.\]

Write the dual support as \(\nu=\lambda^j\zeta h y(h')^3\) and fix \(d,j,\zeta,h,h'\). Put \(D_d=Nd\), \(E=Nh\), \(P'=Nh'\), and \(g=N((\lambda\nu,r))\le\min(R,EP')\). The free squarefree variable has norm length \(Y'\ll I/(D_d^3 3^j E(P')^3)\). The coefficient bounds and the ordinary mean value, after extracting \(\sqrt V\), give \[\ll_{\ell,\epsilon}X^\epsilon R^{-1/2} \frac{3^{-j/6}g}{D_d P'\sqrt E} \left(1+\sqrt{Y'/T}\right).\] Here the free coefficients have size \(O((Ny)^{-1/2})\); collecting equal norms costs only divisor factors. The endpoint \(j=-1\) changes constants. For the first part use \(g\le\sqrt{REP'}\). The \(j\) sum is geometric, the \(d\) sum is logarithmic, the \(h\) choices cost at most \(\tau(r)\), and \(\sum_{h'\mid r^\infty}(Nh')^{-1/2}\ll_\epsilon R^\epsilon\). Its total is \(O(X^\epsilon)\). For the second part use \(g\le EP'\); after substituting \(Y'\) its total is at most \[X^\epsilon\sqrt{\frac I{RT}} \sum_{d,j,h,h'}\frac{3^{-2j/3}} {D_d^{5/2}(P')^{3/2}} \ll X^\epsilon\sqrt{\frac I{RT}} \ll X^\epsilon R^{1/4}\sqrt T.\] The sum over \(h\) is charged by a divisor bound; the other displayed sums converge or cost \(R^\epsilon\). Reselecting the small losses proves (74). ◻

Proof. Nonsquarefree levels vanish. For squarefree \(r\) the exact inverse completion is \[ \sum_u G_\ell(ru)W_r(Nu/U) =\sum_{(c,r)=1}\mu(c)|c|\theta_{3\ell}(c) \mathcal C_{r,\ell}(U/(Nc)^3;W_r). \tag{77}\] Indeed, after \(e=cd\), the angular factor is \(\theta_\ell(rue^3)\) and the divisor coefficient is \(|e|\sum_{c\mid e}\mu(c)\). This is zero unless \(e=1\). In particular, the sign of the angular index on \(c\) is positive.

For low heights take the cutoffs of Proposition 8: \(C_*=X^{.13}\) for \(R\le X^{.40}\) and \(C_*=X^{.02}\) otherwise. The short errors from (73) total \(O_{\ell,\epsilon}(X^\epsilon R^{3/2}C_*^{3/2}) \ll X^{.795+\epsilon}\). When \(\ell=0\) the main term and its tail are exactly those already computed there. When \(\ell\ne0\) the Voronoi main term is absent. The model is small in this case for a separate lattice reason. For a fixed nonzero integer \(\ell\), \[ \sum_{(u,r)=1}\mu^2(u)\theta_\ell(u)W(Nu/U) \ll_{\ell,\epsilon}X^\epsilon\sqrt U. \tag{78}\] To see this, a smooth annular function \(\theta_\ell(z)W(Nz/Y)\) has zero area integral and lattice discrepancy \(O_\ell(L^C(1+\sqrt Y))\), by comparison on fundamental cells and integration of its gradient. Square-divisor and coprimality inversion give \(u=c^2dv\), with \((c,r)=1\) and \(d\mid r\). Rotation contributes only \(\theta_\ell(c^2d)\). For \(Nc\ll\sqrt U\), the constant errors cost \(O(\sqrt U\,\tau(r))\); the radius errors cost at most \(O(\sqrt U\log U)\) times a divisor factor of \(r\). Terms whose scaled support contains no nonzero lattice point vanish. This proves (78). Only the area integral vanishes exactly, not the discrete angular sum.

Apply this estimate to \(x^{-1/6}W_r(x)\). After the outer sum, the model error is \[O_{\ell,\epsilon}(X^\epsilon R^{5/6}U^{1/3}) \ll X^{.589+\epsilon}.\] For the long completion tail, combining \(c,d\) into \(e\) leaves \(\theta_{3\ell}(e)\theta_\ell(r)\theta_\ell(u)\) as independent modulus-one coefficient factors. The same counting bound \(X^{1+\epsilon}E^{-3/2}\) and the same cubic-sieve bound \[X^{1/2+\epsilon} \left(R+U/E^3+(X/E^3)^{2/3}\right)^{1/2}\] therefore apply on \(Ne\asymp E\). The first gives exponent \(.805\) in the small-\(R\) branch; in the other branch the three exponents are \(.755\), \(.77\), and \(5/6-.02\). These, the short errors, and the radial main-term tail when \(\ell=0\) prove (75).

For high heights, let \(b_0\) be a common upper endpoint of the supports of \(W_r\). A nonempty completed sum has \(Nc\le(b_0U)^{1/3}\) and hence \(V=U/(Nc)^3\ge1/b_0\). We may therefore apply Lemma 25 inside the inverse completion. With the norm twist its coefficient is \(\mu(c)|c|\theta_{3\ell}(c)(Nc)^{3it}\). At \(V=U/(Nc)^3\) its error is \(X^\epsilon\sqrt U\,(Nc)^{-1}R^{1/4}\sqrt T\). Only \(Nc\ll U^{1/3}\) can contribute, so this sum is logarithmic. The main term is absent for \(\ell\ne0\). For \(\ell=0\) it is bounded by \(O_D(U^{5/6}R^{-1/6}T^{-D})\) after inversion: the Mellin factor has arbitrary height decay and the \(c\) sum has exponent \(2\). Finally, summing \(|a_r|\) gives the error \(X^{1/2+\epsilon}R^{3/4}\sqrt T\) and the stated pole bound. This proves (76). ◻

Proof. We first prove the version of Lemma 11 needed for a full smooth factor. Let \(\Psi\) be the primitive nonprincipal character inducing \(\chi\Theta_\ell\) in the dominant-full-sum reduction, where \(\chi\) is finite-order cubic, including its fixed local factors. There is no additional imaginary norm parameter: all norm oscillation is the explicit factor \((N\mathfrak n)^{it}\). Mellin inversion gives \[\sum_{\mathfrak n}\Psi(\mathfrak n)(N\mathfrak n)^{it} W(N\mathfrak n/Z) =\frac1{2\pi i}\int \mathcal MW(z)Z^zL(z-it,\Psi)\,dz.\] The sum is over all integral ideals. Move to \(\mathop{\mathrm{Re}}z=-A\) and apply (70). For fixed \(k=|\ell|/2\), the quotient \(\Gamma(1-z+it+k)/\Gamma(z-it+k)\) grows with degree \(1+2A\) in \(1+|\Im z-t|\). Absolute convergence on this line therefore allows truncation into dual norm dyads \[ J\ll_{\ell,\epsilon}Y^\epsilon D(1+|t|)^2/Z, \tag{79}\] with an arbitrarily small power tail. Within each row average fix \(Z,t,W\) and the local and Euler-divisor case, as in the radial Mellin reduction. Choose one cutoff using the largest permitted conductor and the same smooth dyadic partition for every row. This ensures that neither a row-dependent cutoff nor a varying norm shift enters the coefficients.

Move each retained finite dyad to \(\mathop{\mathrm{Re}}z=1/2\). The first numerator gamma pole has real part \(1+k\), so none is crossed. Writing \(z=1/2+i\tau\) and \(C_D=\sqrt{3D}/(2\pi)\), the row-dependent factor is \[\epsilon_\Psi C_D^{2i(t-\tau)} \frac{\Gamma(1/2+k-i(\tau-t))} {\Gamma(1/2+k+i(\tau-t))}.\] It has modulus one. The dyad amplitude is \(\sqrt{Z/J}\), its character is \(\overline\Psi\), and \(|\mathcal MW(1/2+i\tau)|\) supplies a common rapidly decreasing \(L^1\) majorant independent of conductor and height. Its mass is \(O(Y^\epsilon)\) under the stipulated logarithmic derivative bounds. Minkowski’s inequality introduces only the common additional norm shift \(\tau\). On primary dual arguments the angular multiplier is the common coefficient \(\Theta_{-\ell}\).

The local factors require some care. Fix the small twists and the finitely many conductor possibilities above \(3\) as in the dominant-full-sum argument of Section 4. Remove the missing Euler factors before applying the primitive equation. When the primitive character is unramified at \(\lambda\), split its dual \(\lambda\) powers and use \(\overline\Psi(\lambda)^j\). For each fixed \(j\) this is a row scalar of modulus at most one; on the remaining argument the angular coefficient is common. It need not equal the raw element phase \(\theta_{-\ell}(\lambda)^j\). The same finite exclusions and subpower divisor costs as in the radial proof suffice. For \(\ell\ne0\) nonprincipality follows from the infinity type; for \(\ell=0\) it follows from the surviving large-prime part of the row conductor used there.

We now check the entire moment argument. Repeated-prime and prime-power discrepancies have unchanged support and absolute divisor bounds. Thus both discrepancy estimates of Section 4 apply. If \(T_\ell f(n)=\Theta_\ell(n)f(n)\), multiplicativity gives the exact identity \(T_\ell(f*g)=(T_\ell f)*(T_\ell g)\). In particular the short convolution identity used there becomes \[T_\ell\Lambda= \bigl(2T_\ell m-(T_\ell m)*(T_\ell m)*(T_\ell\mathbf1)\bigr) *T_\ell(\log N).\] All truncation ranges and norm dyads are unchanged. In a product of factors, including repeated copies, the merged coefficient is multiplied by \(\Theta_\ell\) of the product argument. The angular order is not multiplied by the number of factors. The resulting coefficients are still common to all cubic-character rows and divisor-bounded.

If a length exponent tends to one, the corresponding full smooth factor is treated by (79). The bound \(|t|\le Y^{.36}\), with the internal allowance \(Y^{.37}\), yields the same dual length \(J\le Y^{.74+o(1)}\) and second-moment exponents \(2.16\) in the first configuration and \(2/3+2(1/3+.37)\) in the second. Both are strictly below \(7/3\). Thus this case is removed exactly with a power gap. Every remaining sieve is applied to the finite cubic row character with \(\Theta_\ell\) in its arbitrary common coefficients.

With full smooth factors excluded, every limiting length exponent is strictly less than one. The remaining moment argument in the proof of Proposition 10 applies to the twisted polynomials: their coefficients are common to all rows and divisor-bounded, and these properties survive the fixed products and repetitions used there. The large-value inequality (27) therefore holds unchanged for a set of \(Y^{r+o(1)}\) rows. In the second configuration the same ordinary-sieve argument, repeating one positive factor, gives \(r\le\max(4/3,z)-2z/3<2/3\), with \(1<z<2\), and the required contradiction.

In the first configuration the multiplicity inequality (28), in the notation of that proof, is still \[r\le\frac23-2\sum_i k_i(v_i-5a_i/6), \qquad \frac12\le\sum_i k_i a_i\le2.\] The radial multiplicity comparison thus produces a polynomial of length \(Z=Y^{z+o(1)}\), with \(4/3<z<2\), whose values are \(Y^{5z/6+o(1)}\) on \(Y^{2/3-o(1)}\) distinct rows. The Gram matrix of Lemma 12 is unchanged: \(\Theta_\ell\) belongs to the common coefficient vector, and cancels against its conjugate in every row product. The radial weight stays radial. Writing \(R\) for the number of these rows, after the same division by \(ZR\) the left exponent is \(2z/3\), whereas the diagonal and off-diagonal exponents are \[z-\frac23,\qquad \frac{z+\max(1,2-2z/3)-2/3}{2}.\] Their gaps are \((2-z)/3\) and, respectively, \((3z-4)/6\) for \(z\le3/2\) or \((z-1)/6\) for \(z\ge3/2\). Each is positive. Choosing the small-power losses after the fixed multiplicity gives the same contradiction. Keep \(\ell\) fixed in the uniform-neighborhood argument at the end of that proof. A sequence of shrinking neighborhoods and savings would again have a convergent subsequence of length and value exponents; the selected multiplicity and losses are fixed before taking the limit. The contradiction therefore gives the required fixed \(\delta,\eta>0\). ◻

Proof. Repeat the largest-prime selection of Lemma 19. Discard the part whose prime factors all have norm at most \(\exp(\sqrt{\log X})\) by its absolute Rankin bound. For the rest, fix all factors except a largest prime \(\pi\), of dyadic norm length \(P\ge\exp(\sqrt{\log X})\), starting this dyadic subdivision at that threshold. Their angular factor is a scalar. The remaining prime sum is tested against \(\chi(\pi)\Theta_\ell(\pi)\), with \(\chi\) nonprincipal cubic of logarithmic conductor, a logarithmic norm height, and the same smooth weights and norm intervals from the stopping rule. Since \((\log X)^C\le(\log P)^{2C}\), the conductor is within Lemma 24. If \(\ell\ne0\) the infinity type prevents principality; if \(\ell=0\) the nonprincipal cubic condition does so. Partial summation supplies arbitrary logarithmic saving uniformly in the remaining norm intervals and heights. The polynomial-size exclusion integer deletes only \(O(\log X)\) possible primes. This costs a logarithmic power per fixed complement, harmless since \(P\ge\exp(\sqrt{\log X})\). Choose the prime saving after the fixed coefficient multiplicities, norm-height powers and desired saving. Summing the complementary factors then gives precisely the bound in (SW). ◻

Proof. Keep \(W_A(a)\) radial and put \[F_{\ell,u}(a)=\sum_b\beta_\ell(b)(Nb)^{iu}G(b) \overline{\chi_b(a)},\qquad S_\ell(u)=\sum_b\beta_\ell(b)(Nb)^{iu-1/6}.\] In every opened pair, the extra factor is \(\Theta_\ell(b_1)\overline{\Theta_\ell(b_2)}\) in the coefficients. At \(b_i=fln_i\) the common factor cancels, leaving the separate twists on \(n_1,n_2\). Poisson summation and its plane transform still act on a radial weight. The diagonal, large-gcd removal, transform tails and all general-coefficient sieves and height means therefore have their original bounds.

At the two exceptional large-conductor configurations the length argument still forces \(f=l=1\), \(B=X^{1/2+o(1)}\), small extra twists and exclusions, and the internal height at most \(B^{.36}\). Proposition 27 supplies their saving. At small nonprincipal conductor use (SW) for \(\beta_\ell\) itself. When \(B\le X^{.40}\) the exceptional configurations remain excluded by the original absolute length gaps. Thus the power-width derivative bounds, sharp norm-cell restrictions and unrestricted translated height in that range retain the same \(\gamma\), independently of \(\ell\).

The cube frequencies give the main term \[ c_*^2|S_\ell(u)|^2 \sum_a\mu^2(a)W_A(a)(Na)^{-1/3}. \tag{80}\] This expression need not vanish for \(\ell\ne0\). Its cancellation uses the corrected square, as in the radial argument. Indeed the mixed term before multiplication by \(\overline{S_\ell(u)}\) is \[\sum_{a,b}\beta_\ell(b)(Nb)^{iu} W_A(a)(Na)^{-1/6}G(ab).\] Here the angular factor is only in the arbitrary outer \(b\)-coefficient. The inner \(a\)-weight is radial, so the original Proposition 8 gives \[c_*\sum_{a,b}\beta_\ell(b)(Nb)^{iu-1/6} \mu^2(ab)W_A(a)(Na)^{-1/3} +O\bigl(A^{-1/6}X^{5/6-\eta}\bigr).\] After multiplication by \(|S_\ell(u)|\ll B^{5/6}L^{O(1)}\), the error is \(O(\mathcal QX^{-\eta}L^{O(1)})\). Roughness and the unchanged absolute coefficient moments remove cross-coprimality at the original logarithmic cost. The positive square, mixed term and model square thus have the same main term (80) with coefficients \(1,-2,1\). This proves the corrected dispersion bound.

In the bilinear sum use \(\alpha_\ell(a)=\Theta_\ell(a)\alpha_0(a)\) and \(\beta_\ell(b)\). The first phase disappears under Cauchy, and its coefficient norm is unchanged. Restoring the model gives exactly \(\theta_\ell(ab)c_*\mu^2(ab)N(ab)^{-1/6}\). All cross-coprimality errors use absolute values. In particular, the three-prime transition still has saving \(L^{-3/2}\).

For the integrated estimate, at large \(B\) the diagonal remains \(A\sqrt B\ll X^{.80}\) and the other term has arbitrary logarithmic saving. At small \(B\) partition both norm variables into the same \(O(J)\) cells. Their coefficient norms \(a_i,b_j\) are unchanged. Near pairs give \(X^{5/6}L^{O(1)}H^{-1/6}J^{-1/2}\) as in (61); all absolute errors sum to \(X^{5/6}L^{O(1)}J^{3/2}\varepsilon_J^{1/2}\) as in (62). For far pairs, integration by parts differentiates the height cutoff. The angular phases stay in the coefficients, and the common Mellin shifts remain independent of the averaging height. Thus the factor \((J/T)^q\) and the unrestricted-shift use of dispersion are preserved. With \(J=L^{D'}\) or \(X^\gamma\) the same positive gaps prove the integrated estimate. ◻

Proof of Theorem 23. Apply Lemma 22 to the prime coefficients with their angular factors and \(T_{\max}=X^{1/6+\rho}\). Equal-norm collection has the same absolute divisor bound, so the truncation error is \(O_\epsilon(X^\epsilon(1+X/T_{\max}))\) and is \(o(X^{5/6}/\log X)\) for \(\epsilon<\rho\). At low heights the exact decomposition just verified, the angular Type I comparison and the angular bilinear comparison give the same arbitrary logarithmic savings. The three-prime transition contributes \(O_\ell(X^{5/6}L^{-3/2}(1+\log L)^C)\), which is also little-oh at the required scale. Sparse terms retain their power savings.

At high heights first bound the undecomposed angular prime model. If \(d_{\ell,n}\) is its coefficient after collecting equal norms, then \(\sum_n|d_{\ell,n}|^2\ll X^{2/3}L^{O(1)}\) by absolute divisor multiplicity. The ordinary mean value therefore gives the same \(T^{-1/2}\) saving used in Section 7. This step does not use angular prime cancellation. For the Gauss-sum term the integrated bilinear estimate is Proposition 29. The high-height Type I estimate gives the same two endpoint powers, \(.805+\epsilon\) and \(5/6-3\kappa/8+\rho/2+\epsilon\). Choose \(\epsilon\) below the two fixed gaps to \(5/6\). Its pole term is absent unless \(\ell=0\), when its arbitrary height decay is the original one. The two-cell and far-pair savings proved above pay for the same polynomial number of pieces and the \(O(L)\) height dyads. Hence the dyadic prime comparison (67) holds with \(\theta_\ell(\pi)\) inserted. Geometric dyadic summation gives (68).

For \(\ell\ne0\), take \(\chi\) principal in Lemma 24 and remove the prime logarithm by partial summation. Thus \(P_\ell(y)=\sum_{N\pi\le y}\theta_\ell(\pi) \ll_{\ell,M}y(\log y)^{-M}\) for every fixed \(M\). Abel summation now gives \[\sum_{N\pi\le X}\theta_\ell(\pi)(N\pi)^{-1/6} =X^{-1/6}P_\ell(X) +\frac16\int_2^X P_\ell(y)y^{-7/6}\,dy =o_\ell(X^{5/6}/\log X).\] The lower endpoint contributes only a bounded term. Combining this with (68) proves (69). ◻