Previous work

Erdős and Pomerance (Erdős and Pomerance 1978) formulated the joint independence problem and proved that each ordering has positive lower natural density. Their explicit lower bound was \(0.0099\). They also proved that for every \(\varepsilon>0\), some \(\delta>0\) makes the number of integers \(n<X\) satisfying \(X^{-\delta}<P^+(n)/P^+(n+1)<X^\delta\) less than \(\varepsilon X\) for all sufficiently large \(X\) (Erdős and Pomerance 1978, Theorem 1). The lower-density bound was increased to \(0.05544\) by de la Bretèche, Pomerance and Tenenbaum (de la Bretèche et al. 2005, sec. 3), and to \(0.05866\) by an observation of Fouvry recorded in the same section. Wang (Wang 2017, 2018) obtained \(0.1063\) and \(0.1356\), and Lü and Wang (Lü and Wang 2025) obtained \(0.2017\). Yang’s recent preprint (Yang 2026, Theorem 1.4) gives \(0.280\) for both orderings. These are bounds on lower natural densities; they do not assert that a natural density exists.

For the joint law itself, Teräväinen (Teräväinen 2018, Theorem 1.14) proved the predicted product in logarithmic density. That is, for fixed \(0<a,b<1\), the indicator in Theorem 1, averaged with weight \(1/n\) and normalization \(1/\log X\), tends to \(\rho(1/a)\rho(1/b)\). His Theorem 1.16 gives logarithmic density \(1/2\) for the ordering, while Theorem 1.19 proves positive lower natural density for every nondegenerate rectangle of normalized largest-prime-factor values inside \((0,1)^2\). Tao and Teräväinen (Tao and Teräväinen 2019, Remark 3.3, equation (50)) then obtained the joint product law for ordinary averages outside an exceptional set of scales of logarithmic density zero; their Corollary 1.16 gives the corresponding ordering result. Wang (Wang 2021) proved the ordinary joint law under the Elliott–Halberstam conjecture for friable integers. Jiang, Lü and Wang (Jiang et al. 2022) established averaged-over-shift forms of the conjectures, a different conclusion from a result at the fixed shift one.

The more recent work of Tao and Teräväinen (Tao and Teräväinen 2026, Theorem 1.8) gives a quantitative joint law outside an exceptional set of scales, with uniformity in the smoothness parameters and a power saving in \(\log X\). Theorem 1 concerns fixed parameters and has no exceptional scales; it does not assert their quantitative error term or their uniformity for growing parameters. Relative to the results just described, the distinction is therefore the unconditional ordinary limit at every sufficiently large scale, not a new marginal distribution or a logarithmically averaged independence statement.

Method and organization

We first replace the largest prime factor by finitely many prime counts. For a fixed integer \(J\ge2\), divide the primes in \((x^{1/J},x]\) into bins \(\mathcal B_{k,x}=(x^{k/J},x^{(k+1)/J}]\), \(1\le k<J\). If \(\Omega_E(n)\) counts prime factors in \(E\) with multiplicity, a choice of unit complex numbers \(\zeta_k\) defines the bin character \[f_x(n)=\prod_{k=1}^{J-1}\zeta_k^{\Omega_{\mathcal B_{k,x}}(n)}.\] Let \(g_x\) be defined by a second arbitrary phase vector, and let \(F_x=f_x-\mu\), where \(\mu\) is the limiting mean of \(f_x\). Finite Fourier inversion reduces the joint law to the mixed decorrelation \[\frac1x\sum_{n<x}\overline{g_x(n)}F_x(n+1)\longrightarrow0.\] Here \(x\) runs through integer scales. The two phase vectors may differ. This freedom gives independence of the two count vectors.

Two properties of these functions connect the beginning and end of the proof. First, every fixed positive integer multiplier has all its prime factors below the bins once \(x\) is large. Thus \(F_x(un)=F_x(n)\) and \(g_x(un)=g_x(n)\) for each fixed \(u\). Second, \(F_x\) has cancellation in averages over growing intervals that remain short relative to the original scale. Let \(B\) be the auxiliary parameter and let \(T=T(B)\to\infty\) be the growing shift scale. If \(P_B\) is a finite set of primes with \(\min P_B\to\infty\), define \[G_{B,v}(n)=\prod_{\substack{p\in P_B\\p\mid n}} \frac{1+p^{-v/B}}2,\qquad v\ge0.\] For any interval \(I_B\) of \(L_B\to\infty\) consecutive integer shifts, and fixed \(0<s_0<s_1\), positive integer \(D\), residue \(r\pmod D\), and \(v\ge0\), Section 2 proves \[\lim_{B\to\infty}\limsup_{x\to\infty} \frac1{Tx}\sum_{s_0Tx\le n\le s_1Tx} \left|\frac1{L_B} \sum_{\substack{i\in I_B\\i\equiv r\pmod D}} F_x(n+i)G_{B,v}(n+i)\right|^2=0.\] At fixed \(B\) the shift interval is short relative to \(x\), while the base points have size \(Tx\). This is the scale produced by the divisor substitution below.

The centered function \(F_x\) is not multiplicative. On the relevant ranges \(n=O_B(x)\), the bin counts take values in a fixed finite set. This permits interpolation by a fixed finite family of real, nonnegative multiplicative functions. After resolving the residue condition by Dirichlet characters, the real short-interval theorem of Matomäki and Radziwiłł (Matomäki and Radziwiłł 2016, Theorem 1) compares the principal components with long means whose centered combination vanishes. The complex theorem of Matomäki, Radziwiłł and Tao (Matomäki et al. 2015, Theorem A.1, corrected version) controls the nonprincipal twists. The rest of the proof must reduce the mixed correlation to this particular cancellation statement.

Suppose, then, that the mixed correlation stays nonzero along a sequence of scales. The profinite integers \(\widehat{\mathbb Z}\) encode compatible residue classes modulo all positive integers and carry Haar probability measure. Passing to a subsequence gives a bounded profile \(W(t,w)\) on \((0,\infty)\times\widehat{\mathbb Z}\). Its integral against each fixed compactly supported continuous test \(\Phi(t,w)\) is the subsequential limit of the mixed averages weighted by \(\Phi(n/x,n)\), with \(n\) in the second argument viewed through its residues. A fixed nonnegative compactly supported smooth cutoff \(\phi\) can be chosen so that \(\int\phi(t)W(t,w)\,dt\,dw\ne0\), where \(dt\) is Lebesgue measure and \(dw\) is Haar measure.

For an integer \(n\), the amplifier defines \(D_B(n)\) as a nonnegative weighted count of factorizations \(n=am\) and \(n+1=cl_a\), restricted by \(e^B<c<e^{2B}\) and \(Tc<a<2Tc\). At fixed \(B\) its coefficient list is finite, and the same formula defines a function on \(\widehat{\mathbb Z}\) depending only on residues at primes above a cutoff tending to infinity. The construction gives \(D_B\) Haar mean at least \(d_0>0\) for large \(B\) and bounded \(L^2\) norm. To prove these bounds, the argument passes to a comparison model in which each site prime is assigned by an independent fair coin to the coefficient divisor or the remaining factor. The first and second moments are analyzed through one split and two conditionally independent splits of the same site prime sets. For each fixed \(B\), \(\Phi(t,w)=\phi(t)D_B(w)\) is an allowed profile test, so along the selected subsequence \[\lim_{x\to\infty}\frac1x\sum_{n\ge1} \phi(n/x)\overline{g_x(n)}F_x(n+1)D_B(n) =\int\phi(t)W(t,w)D_B(w)\,dt\,dw.\] Independence from every fixed residue coordinate and \(L^2\) approximation of \(\phi W\) show that the right-hand side has absolute value bounded away from zero as \(B\to\infty\).

For fixed \(B\) and large \(x\), writing \(n=am\) in this weighted average replaces \(g_x(n)\) by \(g_x(m)\). For each fixed pair \((c,m)\), the remaining terms form a sum over \(a\) satisfying \(c\mid am+1\), with nonnegative divisor and smooth weights multiplying \(F_x(am+1)\). Cauchy–Schwarz in \((c,m)\) removes the unit-modulus factor \(g_x(m)\) and squares this inner sum. The resulting normalized quadratic energy has a positive lower bound, while its equal-coefficient diagonal tends to zero in the same iterated limit. A positive contribution therefore remains from distinct indices \(a,b\). Section 4 proves these moment and energy claims.

For such a pair, \(c\) divides both \(am+1\) and \(bm+1\). Thus \(m\) is a unit modulo \(c\) and \(a-b=jc\), with \(0<|j|<T\) by the coefficient windows. Writing \(l_a=(am+1)/c\) and \(l_b=(bm+1)/c\), the exact substitution \[n'=b l_a,\qquad n'+j=a l_b\] turns the label product into \(\overline{F_x(n')}F_x(n'+j)\) by invariance under the fixed multipliers \(a,b,c\). The new endpoints have size \(Tx\). The off-diagonal energy is therefore a weighted graph of ordinary additive shifts. Its edge weights depend on endpoint prime sets and on additional residue data from each divisor representation.

Group the new endpoints into blocks \(N+i\), \(1\le i\le M\), with \(M=\lceil C_6T\rceil\) for a fixed large \(C_6\), and put \(s=N/(Tx)\). Section 5 couples the actual auxiliary-prime divisibility sets at these positions to independent model sets \(S_i\), each formed by including every auxiliary prime \(p\) independently with probability \(1/p\), and averages the extra residue weight of each representation. This compares the arithmetic graph with an endpoint kernel \(\mathcal L_{ik}(S_i,S_k;s)\) in expected cut norm. Here cut norm is the largest absolute normalized pairing with two real vectors bounded by one, chosen after the matrix is known. Section 6 controls representation multiplicity and provides the averaged second-moment bounds needed for sampling.

The aim is now to replace this endpoint kernel by products of functions of its two endpoints. For an endpoint type \(S\), let \(b\) be the product obtained by retaining each prime independently with probability \(1/2\). The features used in the comparison are \[V_l(S)=\frac1{|I_l|} \mathbb P_{\rm split}\!\left(\frac{\log b}{B}\in I_l\,\middle|\,S\right),\] where the \(I_l\) form a fixed coarse partition of a compact logarithmic interval, chosen for a desired accuracy. The channel estimates in Section 7 show that fair splitting suppresses residue dependence and makes logarithmic outputs uniformly approximable on this partition. Fourier detection of \(a-b=jc\) then lets Section 8 apply arithmetic estimates to the coefficient \(c\) and these channel estimates to the endpoints. The resulting comparison has the form \[\sum_l c_{l,B}(j,s)V_l(S_i)V_l(S_k),\qquad j=k-i,\] where \(i,k\) are the endpoint positions. The scalar coefficients separate an arithmetic factor in \(|j|\) from a smooth factor in the normalized lag \(|j|/T\) and the common block origin \(s\). The number of features is fixed, while their values and the coefficients may depend on \(B\).

The feature comparison first holds after integration against separate functions of the two independent endpoint sets. The actual labels need not have this single-site form, and an optimizing test can depend on the whole configuration. Section 9 upgrades the comparison to expected cut norm by approximating an optimizing cut with a small sample of columns. Finally, polynomial approximation of each log-window indicator expresses \(V_l\), up to a small mean-square error, as a fixed finite combination of \[\mathbb E_{\rm split}\!\left[e^{-v\log b/B}\mid S\right] =\prod_{p\in S}\frac{1+p^{-v/B}}2.\] At an integer endpoint these are exactly \(G_{B,v}\). The arithmetic lag factor is approximated in mean over lags by a fixed periodic function. For a fixed small \(\delta>0\) chosen for the desired accuracy, subdivide each position block into a fixed number of intervals whose lengths grow with \(T\) and are at most \(\delta T\). Within each interval pair, the normalized lag varies by at most \(2\delta\). Approximating the smooth lag factor there and resolving the periodic factor into fixed residue classes turns the feature energy, up to a controlled error, into products of the weighted short averages already controlled in Section 2. Their vanishing contradicts the positive energy. Sections 10 and 11 complete this argument and the passage from bin events to the joint law.

The divisor amplification and graph comparison have close antecedents in Tao’s logarithmic correlation argument (Tao 2016), the prime-divisibility graphs of Helfgott and Radziwiłł (Helfgott and Radziwiłł 2021), Pilatte’s product-of-primes amplification (Pilatte 2023), and the mixed-function decoupling of Tao and Teräväinen (Tao and Teräväinen 2026, sec. 3.1). The present rough-divisor graph and its endpoint comparison are proved locally. The sampling argument likewise builds on the cut-based methods of Frieze and Kannan (Frieze and Kannan 1999), Alon, Fernandez de la Vega, Kannan and Karpinski (Alon et al. 2003, Lemma 3), and Borgs, Chayes, Lovász, Sós and Vesztergombi (Borgs et al. 2008, Theorem 4.6). Because the kernels here vary with \(B\) and can be large, the proof establishes the moment bounds needed before applying bounded-differences concentration (McDiarmid 1989).

The arithmetic estimates are developed in Section 3 using Selberg–Delange theory, character estimates and upper-bound sieves in the forms recorded by Granville and Koukoulopoulos (Granville and Koukoulopoulos 2019), Koukoulopoulos (Koukoulopoulos 2019), and Ford (Ford 2023, 2002). The minor-arc input in Section 8 is the exponential-sum estimate of Montgomery and Vaughan (Montgomery and Vaughan 1977). Throughout the proof, the original scale \(x\) tends to infinity before the auxiliary scale \(B\). Bin data, feature degrees, residue moduli and divisor multipliers are fixed in that inner limit. The contradiction applies to a subsequence of every possible bad sequence of original scales, and hence yields the full ordinary limit.

Labels and their one-variable distributions

Fix an integer \(J\geq2\). For \(1\leq k<J\) put \[ \mathcal B_{k,x}=(x^{k/J},x^{(k+1)/J}]. \tag{1}\] If \(E\) is a set of primes, \(\Omega_E(n)\) denotes the number of prime factors of \(n\) belonging to \(E\), counted with multiplicity. Fix complex numbers \(\zeta_1,\ldots,\zeta_{J-1}\) and \(\xi_1,\ldots,\xi_{J-1}\) of absolute value one, independent of \(x\), with no relation required between the two vectors. Define the completely multiplicative function \(f_x\) by \[ f_x(p)= \begin{cases} \zeta_k,&p\in\mathcal B_{k,x},\quad 1\leq k<J,\\ 1,&p\notin\bigcup_{1\leq k<J}\mathcal B_{k,x}. \end{cases} \qquad f_x(n)=\prod_{k=1}^{J-1}\zeta_k^{\Omega_{\mathcal B_{k,x}}(n)}. \tag{2}\] Define the completely multiplicative label associated with \((\xi_k)\) by \[ g_x(n)=\prod_{k=1}^{J-1}\xi_k^{\Omega_{\mathcal B_{k,x}}(n)}. \tag{3}\] Both labels have absolute value one. For every fixed \(C>0\), every integer \(n\leq Cx\) has at most \(J\) prime factors in the union of the bins, once \(x\) is sufficiently large in terms of \(C,J\). Indeed, \(J+1\) such factors would have product exceeding \(x^{1+1/J}>Cx\).

The centered function \(F_x\) is not asserted to be multiplicative. Its eventual invariance under each fixed multiplier is the exact property in (6) that will be used below.

The same distributional limit holds for averages over \(1\leq n\leq x\). Indeed, apply the lemma to any fixed function of the count vector on \(\alpha x<n\leq x\), then let \(\alpha\downarrow0\); that function is bounded on the finite set of possible vectors, so the omitted interval has normalized contribution \(O(\alpha)\). In particular, the limit in (9) is unchanged when its normalized progression sum is replaced by \(x^{-1}\sum_{1\le n\le x}\prod_k(\Omega_{\mathcal B_{k,x}}(n))_{r_k}\). This gives (4) on the initial segment as well. A fixed shift of the integer argument changes only a bounded number of terms after nonpositive arguments are omitted, so the label means have the same limits after such a shift.

The mixed correlation and its short-average input

The joint law will follow by finite Fourier inversion from the following statement for the two independently chosen phase vectors. The factor \(F_x(n+1)\) is centered, while \(g_x(n)\) has absolute value one.

After establishing the short-average input below, we will suppose that the average in (10) stays a fixed positive distance from zero along a sequence and derive a contradiction. The input is used in Section 10 at the final step of the proof.

For each auxiliary parameter \(B\), let \(P_B\) be a finite set of primes, and suppose that \(\min P_B\to\infty\) as \(B\to\infty\). An empty set may be allowed by interpreting its minimum as \(+\infty\) and its product as one. For a fixed real \(v\geq0\), define \[ G_{B,v}(m)=\prod_{\substack{p\mid m\\p\in P_B}} \left(\frac12+\frac12p^{-v/B}\right). \tag{11}\] This is a real, nonnegative, 1-bounded multiplicative function. For fixed \(B\) it is periodic, with period dividing \(\prod_{p\in P_B}p\). To see its fair-split meaning, put \(S_B(m)=\{p\in P_B:p\mid m\}\) and form a random product \(b\) by including each prime of \(S_B(m)\) independently with probability \(1/2\). The empty product is one. Independence gives \[G_{B,v}(m) =\mathbb E_{\mathrm{split}}\!\left[\exp\!\left(-\frac{v\log b}{B}\right)\right].\] These are the fair-split averages used to approximate the endpoint features in Section 10.

The proof interpolates the centered function of the finitely many bin counts by real, nonnegative multiplicative functions. Once the residue of the origin \(n\) is fixed, the congruence on \(i\) becomes a fixed residue condition on the argument \(m=n+i\). Resolving that condition by Dirichlet characters produces two different tasks. For the principal character, short averages must be compared with long means whose centered linear combination vanishes. For a nonprincipal character, the short averages themselves must be small. We record the two published inputs for these tasks and prove the required distance estimate before returning to the weighted lemma.

The short-interval inputs

For a bounded arithmetic function \(g\) and \(H>0\), set \[\mathcal A_Hg(z)=\frac1H\sum_{z<n\leq z+H}g(n), \qquad \mathcal L_Xg=\frac1X\sum_{X<n\leq2X}g(n).\] For a 1-bounded multiplicative function \(g\), define \[ \mathbb D(g,n^{it};X)^2 =\sum_{p\leq X}\frac{1-\operatorname{Re}(g(p)p^{-it})}{p}, \qquad M(g;X)=\inf_{|t|\leq X}\mathbb D(g,n^{it};X)^2. \tag{13}\]

We use the following two forms of the short-interval theorems. Their uniformity in the multiplicative function is part of the statements.

This is a consequence of the uniform exceptional-set estimate in (Matomäki and Radziwiłł 2016, Theorem 1), using boundedness to pass to mean square. The terms in that estimate depending on \(X\) can be absorbed into \(r(H)\) because \(X\geq H\).

This is (Matomäki et al. 2015, Theorem A.1), in the revised version with the corrected proof of Proposition A.3. That proof permits the factor \(e^{-M}\); we use the weaker \((1+M)e^{-M}\), which also covers \(M<1\). Thus divergence of the minimum over \(|t|\leq X\) suffices for our application.

The preceding statements also hold with integer origins, at the cost of an error tending to zero with \(H\). Indeed, for \(|g|\leq1\), \[|\mathcal A_Hg(z)-\mathcal A_Hg(\lfloor z\rfloor)|\leq\frac2H.\] Integrating over unit intervals proves the claim, with \(O(1/X)\) errors at the endpoints. Changing a short interval endpoint by a bounded amount similarly costs \(O(1/H)\).

A uniform estimate for the character twists

Proof of the weighted short-average lemma

A profile of a nonzero mixed correlation

We begin the proof by supposing that Proposition 4 fails. Fix an offending \(J\) and pair of phase vectors. Boundedness permits passing to a subsequence on which the mixed correlation has a nonzero limit: \[ \lim_{x\to\infty} \frac1x\sum_{1\leq n<x} \overline{g_x(n)}F_x(n+1)\ne0. \tag{20}\] The arithmetic amplification in the following sections will contradict this limit; Section 10 completes the argument.

Let \(\widehat{\mathbb Z}\) be the profinite integers, with Haar probability measure \(dw\), and identify every integer with its natural image in \(\widehat{\mathbb Z}\). On \((0,\infty)\times\widehat{\mathbb Z}\) define the locally finite measures \[\lambda_x=\frac1x\sum_{n\ge1}\delta_{(n/x,n)},\qquad \nu_x=\frac1x\sum_{n\ge1} \overline{g_x(n)}F_x(n+1)\delta_{(n/x,n)}.\] Unweighted scale counting gives \(\lambda_x\to dt\,dw\) against compactly supported continuous tests. This follows first for a continuous test in \(t\) times a residue-class indicator in \(w\) by progression counting, and then for every compactly supported continuous test by uniform approximation. Also \(|\nu_x|\le2\lambda_x\). The resulting uniform variation bounds on compact sets, weak compactness on a countable exhaustion, and a diagonal extraction give a further subsequence along which \(\nu_x\) converges against these tests to a locally finite complex measure \(\nu\).

For every compactly supported continuous \(\Phi\), the weak convergences give \[\left|\int\Phi\,d\nu\right| =\lim_{x\to\infty}\left|\int\Phi\,d\nu_x\right| \le2\lim_{x\to\infty}\int|\Phi|\,d\lambda_x =2\int_0^\infty\int_{\widehat{\mathbb Z}}|\Phi(t,w)|\,dw\,dt.\] The dual characterization of variation implies \(|\nu|\le2\,dt\,dw\). The Radon–Nikodym theorem therefore gives a measurable function \[ W:(0,\infty)\times\widehat{\mathbb Z}\longrightarrow\mathbb C, \qquad |W(t,w)|\leq2\quad\text{almost everywhere}, \tag{21}\] such that \[ \lim_{x\to\infty}\frac1x\sum_{n\geq1} \overline{g_x(n)}F_x(n+1)\Phi(n/x,n) =\int_0^\infty\int_{\widehat{\mathbb Z}}\Phi(t,w)W(t,w)\,dw\,dt \tag{22}\] for every compactly supported continuous \(\Phi\) on the product. Here and henceforth limits in \(x\) may use the fixed subsequence.

Boundedness also permits replacing the compactly supported continuous tests in (22) by the indicator of \(0<t<1\): truncate near \(0\), approximate the remaining interval at its endpoints, and use the uniform bound on both weighted counting measures and \(W\). Equation (20) therefore implies \[\int_0^1\int_{\widehat{\mathbb Z}}W(t,w)\,dw\,dt\ne0.\] Approximating this interval indicator by nonnegative smooth functions supported in \((0,1)\), we can fix a real nonnegative \(\phi\in C_c^\infty((0,\infty))\) with \[ \beta_*= \left|\int_0^\infty\int_{\widehat{\mathbb Z}} \phi(t)W(t,w)\,dw\,dt\right|>0. \tag{23}\] The rest of the proof derives a contradiction from this fixed profile and bump. Every auxiliary choice will be made after \(J,f_x,g_x,F_x,W,\phi\) and the subsequence have been fixed.

Uniform local asymptotics

Our local estimates have two outputs. For \(A_0\), they give a smooth mean on multiplicative windows at log scale \(B\), including additive phases with moduli and real frequencies bounded by fixed powers of \(B\). For a product \(a\) formed by selecting each \(p\in\mathcal P\) independently with probability \(z/p\), where \(z\in\{1/4,1/2\}\), they give a smooth density for \(\log a/B\) after restriction to a unit residue class, with an absolute error useful even on intervals of width \(O(1/B)\) in that coordinate. Both outputs follow by partial summation from cumulative estimates with a saving of any prescribed power of \(B\). We first recall the unrestricted Selberg–Delange estimates, then exclude the primes at most \(P_0\) uniformly as that cutoff moves. For these two values of \(z\), define \[g_z(n)=\mu_{\rm Mob}^2(n)z^{\omega(n)},\qquad g_{z,P_0}(n)=g_z(n)\mathbf 1_{n\ {\rm rough}}.\]

We now turn these counting estimates into local densities for the coefficient weights and probability laws for randomly selected prime products. Choose the expansion order once, sufficiently large to use Lemma 10 with \(D=200\) for both values of \(z\). For \(L>0\) write \[P_{z,B}(L)=\sum_{j=0}^{H}C_{j,z}(P_0)L^{z-1-j},\qquad D_{z,B}(L)=P_{z,B}(L)+P'_{z,B}(L).\] Thus \(D_{z,B}(\log Y)\) is exactly the derivative with respect to \(Y\) of \(YP_{z,B}(\log Y)\). Define \[m_B(s)=(\log P_0)R^{1/2}D_{1/2,B}(Bs),\qquad Q_{z,B}=\prod_{p\in\mathcal P}(1-z/p),\qquad f_{z,B}(s)=B Q_{z,B}D_{z,B}(Bs).\] These are finite combinations of smooth powers on \(s>0\). The coefficient bounds and \(\log P_0=B^{o(1)}\) imply, for every fixed nonnegative derivative order, convergence of these functions and their derivatives uniformly on compact subintervals of the indicated domains: \[m_B(s)\longrightarrow m(s)=c_A s^{-1/2}\quad(0<s<4),\qquad f_{z,B}(s)\longrightarrow c_zs^{z-1}\quad(0<s<3.3),\] where \[c_A=\frac{e^{-\gamma_{\rm E}/2}}{\Gamma(1/2)},\qquad c_z=\frac{e^{-\gamma_{\rm E}z}}{4^z\Gamma(z)}.\] Here \(Q_{z,B}\sim(4R)^{-z}\) by Mertens’ estimate. To check the derivative assertion directly, the \(j=0\) term has the stated limit after normalization. Every term with \(j\ge1\), as well as the \(P'_{z,B}\) part, gains at least one power of \(B^{-1}\) relative to that term on a fixed compact interval, with only a fixed power of \(\log P_0\) lost. The same reasoning applies after any fixed number of \(s\)-derivatives. Moreover the leading term dominates uniformly for \(Bs\ge B^{0.89}\): each ratio of a lower-order term to it is bounded by a fixed power of \(\log P_0\) divided by a positive power of \(Bs\). Consequently \(f_{z,B}(s)\ll s^{z-1}\) for \(B^{-1/10}\le s\le3.2\), and all these densities are positive on that range for sufficiently large \(B\).

Upper sieve bounds with reducing local weights

We state precisely the classical upper-sieve input, in its event-space form. For a fixed positive integer \(k\), choose once a bound \(P_k>2k\) depending only on \(k\). For a sieve bound \(Z\ge2\), let \(\mathcal R_Z\subset\{p:P_k<p\le Z\}\) be the set of retained primes. Suppose their forbidden local densities obey \(0\le\theta_p\le k/p\). Mertens’ theorem then gives the upper dimension condition \[\prod_{\substack{p\in\mathcal R_Z\\y<p\le z}}(1-\theta_p)^{-1} \le\left(\frac{\log z}{\log y}\right)^k \left(1+O_k(1/\log y)\right)\qquad(2\le y\le z).\] Let \(\mathcal V\) be the ambient mass. For each squarefree \(d\) whose prime factors all lie in \(\mathcal R_Z\), let \(\mathcal V_d\) be the mass satisfying the forbidden condition at every prime dividing \(d\), and put \[r_d=\mathcal V_d-\mathcal V\prod_{p\mid d}\theta_p.\] Extend \(r_d\) by zero to all other positive integers. The classical fundamental lemma of the upper sieve supplies \(s_k>0\), depending only on \(k\), and upper weights of absolute value at most one, supported on squarefree \(d\le D\) with prime factors in \(\mathcal R_Z\), such that, when \(Z\le D^{1/s_k}\), the mass avoiding the retained forbidden conditions is at most \[C_k\mathcal V\prod_{p\in\mathcal R_Z}(1-\theta_p) +\sum_{d\le D}|r_d|.\] The constants are uniform over the forbidden sets satisfying the displayed dimension condition. This is the bounded-dimension upper-sieve statement of (Ford 2023, Theorems 2.4 and 3.6).

In the remaining estimates, \(n\asymp X\) and analogous notation mean membership in an interval between fixed positive constant multiples of the indicated scale. Restricting to additional size or positivity conditions only decreases the upper bounds. They are uniform when \((\log X)/B\) lies in any fixed compact subinterval of \((0,4)\), with constants allowed to depend on that subinterval. In particular, they apply throughout \(0.9B\le\log X\le2.2B\), the range used later. In applications involving \(TX\), this convention still leaves all coefficient logarithms below \(4B\).

Fix a sufficiently large absolute constant \(C_2\) and define, for nonzero integers \(j\), \[\Sigma(j)=\prod_{p\mid j}(1+C_2/p).\] For every fixed \(r>0\) this function satisfies \(\sum_{1\le j\le U}\Sigma(j)^r\ll_r U\) for \(U\ge1\). Indeed expand \(\Sigma(j)^r=\sum_{d\mid j}b_r(d)\), with \(b_r\) nonnegative, supported on squarefree integers, and \(b_r(p)=(1+C_2/p)^r-1=O_r(1/p)\). Then \(\sum_d b_r(d)/d=\prod_p(1+b_r(p)/p)<\infty\), proving the assertion by summing the divisor expansion.

Regularity cutoffs and their loss

Fix an integer \(L\ge1\) and let \(\Delta=1/L\). The grid consists of \(g=i/L\), \(1\le i\le L\). Fix a small \(\tau>0\) and, later, a sufficiently large \(C_*>0\). The choices of \(L\) and \(\tau\) remain available for the additional fixed requirements in Section 6: first take \(L\) sufficiently large, then take \(\tau\) sufficiently small. None of the estimates of the present section requires an upper bound for \(L\) or a positive lower bound for \(\tau\).

The prefix bounds will control competing divisor representations in Section 6; the tail lower bounds will make the two-split second moment in Section 4 summable.

Put \(A=A_0\mathbf 1_{\rm reg}\) and \(K=K_0\mathbf 1_{\rm reg}\), with the analogous definition for \(K(S)\). The lower total-count bound in the definition gives \[ A(a),\ K(v),\ K(S)\le B^{h_0+\tau\log2+o(1)},\qquad h_0=\frac{1-\log2}{2}. \tag{32}\] For \(A\), any prime factors outside \(\mathcal P\) only decrease the weight; the additional factor \(\log P_0\) is \(B^{o(1)}\). Choose \(\tau\) sufficiently small that \[ 2(h_0+\tau\log2)<0.32,\qquad 1-6(h_0+\tau\log2)>0.07,\qquad 1-5(h_0+\tau\log2)>0.229. \tag{33}\] These are compatible strict conditions, since \[2h_0=0.3068528\ldots,\quad 1-6h_0=0.0794415\ldots,\quad 1-5h_0=0.2328679\ldots.\] Any further decrease of \(\tau\) preserves them.

The divisor weight and the correlation it preserves

For an integer \(n\ge1\), consider the two factorizations \[n=am,\qquad n+1=cl.\] We select \(c\) with \(\log c/B\) in the support of \(\rho_0\) and \(a\) with \(a/(Tc)\) in the support of \(\psi\). The weights \(A(a),A(c)\) apply to the chosen divisors and \(K(m),K(l)\) to the two quotients. Define their normalized divisor sum on the profinite integers by \[ D_B(w)=\frac1B \sum_{\substack{a\mid w\\ c\mid w+1}} A(a)K(w/a)A(c)K((w+1)/c) \rho_0\left(\frac{\log c}{B}\right) \psi\left(\frac{a}{Tc}\right), \qquad w\in\widehat{\mathbb Z}. \tag{35}\] For integer \(w=n\), this is exactly the sum over the two factorizations above. In \(\widehat{\mathbb Z}\), divisibility means membership in the image of multiplication by the divisor. Multiplication by a positive integer is injective on \(\widehat{\mathbb Z}\), so each quotient in the sum is well defined.

For fixed \(B\), the coefficient support is finite: \(e^B<c<e^{2B}\) and \(Tc<a<2Tc\). Every coefficient is less than \(e^{4B}\) for sufficiently large \(B\), so every prime of a nonzero coefficient weight belongs to \(\mathcal P\). The modulus \(\prod_{p\in\mathcal P}p^2\) suffices to determine \(D_B\): a coefficient is squarefree, and testing whether \(p\) divides its quotient requires at most the residue modulo \(p^2\). Thus \(D_B\) is a nonnegative function of finitely many high-prime residues.

Define \(I_{1,x}\) by the weighted correlation \[ \frac{I_{1,x}}B =\frac1x\sum_{n\ge1} \phi(n/x)\overline{g_x(n)}F_x(n+1)D_B(n). \tag{36}\] For each fixed \(B\), the function \(\phi(t)D_B(w)\) is a compactly supported continuous profile test. The convergence in (22) therefore gives \[ \lim_{x\to\infty}\frac{I_{1,x}}B =\int_{(0,\infty)\times\widehat{\mathbb Z}} \phi(t)W(t,w)D_B(w)\,dt\,dw. \tag{37}\]

The factor \(1/B\) in (35) is explained by the fair-split identity (24). Each finite prime set contributes a factor \(B\) when its divisors are summed as a fair split. For two independent products with law \(\nu_{1/2}\), the first-moment calculation below will show that the smooth size-and-ratio expectation is a positive constant divided by \(B\), up to a smaller error. The two factors \(B\) and the external division by \(B\) then give a bounded positive mean in that model. The next reduction compares this model with the actual divisor weight and bounds the exceptional cases.

The moment target and the independent split model

We prove this lemma after reducing the actual divisor weight to an independent model and establishing the concentration estimate needed for its second moment. A single split supplies the positive mean. Two splits of the same prime sets govern the second moment.

Let \(S_1,S_2\) be independent random subsets of \(\mathcal P\), each formed by including every prime \(p\) independently with probability \(1/p\). Conditional on these sets, split each \(S_\nu\) by independent fair coins into a coefficient subset \(\mathcal C_\nu\) and a remaining subset \(\mathcal R_\nu\). Write \[a=\prod_{p\in\mathcal C_1}p,\qquad c=\prod_{p\in\mathcal C_2}p,\] with empty products equal to one, and define \[ \widetilde D_B(S_1,S_2) =B\,\mathbb E_{\mathrm{split}}\left[ \rho_0\left(\frac{\log c}{B}\right) \psi\left(\frac{a}{Tc}\right) \mathbf 1_{\text{all four subsets are regular}} \,\middle|\,S_1,S_2\right]. \tag{39}\] In particular, \(0\le\widetilde D_B\ll B\) pointwise.

The two-split question and an addition-product estimate

The second moment of (39) involves two conditionally independent fair splits of the same two site sets. Denote their coefficient products by \((a^{(h)},c^{(h)})\), \(h=1,2\), and their coefficient and remaining prime sets by \(\mathcal C_\nu^{(h)},\mathcal R_\nu^{(h)}\), \(\nu=1,2\). Choose fixed closed intervals \(I_{\rho_0}\subset(1,2)\) and \(I_\psi\subset(1,2)\) containing the supports of \(\rho_0\) and \(\psi\), respectively. Let \(\mathcal H_h\) be the size-and-ratio event \[\frac{\log c^{(h)}}B\in I_{\rho_0}, \qquad \frac{a^{(h)}}{Tc^{(h)}}\in I_\psi,\] and let \(\mathcal E_h\) be \(\mathcal H_h\) together with regularity of its four subsets. Since the smooth weights are bounded and nonnegative, \[ \mathbb E\widetilde D_B^2 \ll B^2\mathbb P(\mathcal E_1\cap\mathcal E_2). \tag{40}\] It therefore suffices to prove the global probability bound \(\mathbb P(\mathcal E_1\cap\mathcal E_2)\ll_{C_*}B^{-2}\).

To prove this target, we will group the changes between the two splits by their largest logarithmic scale \(Y\). A further unconditional coupling will replace the first coefficient and remaining classes by independent prime processes, with its error paid before conditioning. In that comparison model, the additions from a remaining set at primes with \(\log p\le Y\) are independent selections with probabilities \(1/(4p)\), independent also of the data above \(Y\). After fixing the retained coefficient primes and the additions at the other site, the second ratio window confines the logarithm of the recipient’s addition product to an interval of fixed length. A new prime with logarithm above \(Y/2\) also forces that product logarithm above \(Y/2\). The next estimate gives the resulting \(O(1/Y)\) probability uniformly in the interval’s location. The second-moment proof will combine it with the retained first-remainder tail restrictions and the high assignment coins. The product itself need not be bounded by the largest allowed prime.

The first and second moments

The correlation survives the divisor weights

The factor \(\int D_B\,dw\) in (41) is real, nonnegative, and at least \(d_0\) eventually. Taking complex absolute values in (37) and using (41) therefore gives \[ \liminf_{B\to\infty}\ \lim_{x\to\infty} \frac{|I_{1,x}|}{B}\ge d_0\beta_*. \tag{42}\] The possible dependence of the \(L^2\) constant on \(C_*\) is harmless: this approximation is performed with \(C_*\) fixed, and the positive lower constant in (42) is independent of it.

A positive quadratic energy and its diagonal

The preserved correlation now has an expansion suited to Cauchy–Schwarz. For \(c\mid am+1\), put \(l_a=(am+1)/c\). At fixed \(B\), the coefficient list in (35) is finite. Thus \(g_x(am)=g_x(m)\) holds simultaneously for every coefficient \(a\) on that list once \(x\) is sufficiently large, by (6). Expanding (36), writing \(n=am\), and using this fixed-multiplier invariance gives the exact identity \[ I_{1,x}=\frac1x\sum_{c\ge1} \rho_0\left(\frac{\log c}{B}\right)A(c) \sum_{m\ge1}K(m)\overline{g_x(m)} \sum_{\substack{a\ge1\\ c\mid am+1}} A(a)K(l_a)\psi\left(\frac{a}{Tc}\right) \phi\left(\frac{am}{x}\right)F_x(am+1). \tag{43}\] The values of \(F_x\) are unchanged in this reindexing. The expanded form places \(g_x(m)\) outside the inner sum, where \(|g_x(m)|=1\) supplies the unit factor in Cauchy–Schwarz.

The change of variables

Consider distinct coefficients \(a,b\) in the inner square defining \(I_{2,x}\), for fixed \(m,c\), and put \(l_a=(am+1)/c\) and \(l_b=(bm+1)/c\). The congruences imply that \(m\) is a unit modulo \(c\) and that \[a-b=jc,\qquad 0<|j|\le T.\] They also imply \((a,c)=(b,c)=1\). Since \(a,b\) are rough and \(|j|\le T<P_0\), we have \((a,j)=(b,j)=1\), and hence \((a,b)=1\). For the ordered term in which the \(a\)-factor is conjugated, set \[n'=b l_a,\qquad n'+j=a l_b.\] For these fixed coefficients, this is a bijective reindexing by the conditions \(b\mid n'\) and \(a\mid n'+j\). Indeed, write \(n'=bz\). The latter congruence, together with \(b=a-jc\), gives \(j(1-cz)\equiv0\pmod a\). Thus \[m=\frac{cz-1}{a}\] is an integer, and substitution recovers both original congruences and both displayed identities. Positive compact scale supports ensure all arguments are positive for large \(x\).

Figure 1 records the two fixed multiplications that produce the additive shift. Only fixed-multiplier invariance is used; the centered function \(F_x\) need not be multiplicative.

For fixed \(B\), the coefficient supports are finite, so (6) applies simultaneously to the multipliers \(c,b,a\) once \(x\) is sufficiently large. Removing \(c\) and then inserting \(b,a\) gives \[\overline{F_x(am+1)}F_x(bm+1) =\overline{F_x(l_a)}F_x(l_b) =\overline{F_x(n')}F_x(n'+j).\] Define \[\Psi_s(u_1,u_2)= \psi(u_1)\psi(u_2)\phi(s/u_1)\phi(s/u_2).\] If \(s=n'/(Tx)\), \(u_1=a/(Tc)\) and \(u_2=b/(Tc)\), then \[s/u_1=bm/x+b/(ax),\qquad s/u_2=am/x+1/x.\] The two original \(\phi\)-arguments are therefore interchanged, with errors \(O_B(1/x)\). Since they occur as a product, their replacement by \(\Psi_s\) has vanishing error for fixed \(B\). There is a fixed compact interval \(\mathcal S\subset(0,\infty)\) outside which \(\Psi_s\) vanishes on the coefficient supports. It depends only on \(\phi,\psi\).

Consequently, the off-diagonal part of \(I_{2,x}/(BT)\) is, up to \(o_x(1)\), \[ \frac1{Tx}\sum_{n'}\sum_{0<|j|\le T} \overline{F_x(n')}F_x(n'+j) \mathcal K_j(n',n'/(Tx)), \tag{45}\] where, for \(u\in\widehat{\mathbb Z}\), \[ \begin{split} \mathcal K_j(u,s)=\frac1B \sum_{\substack{a-b=jc,\ (a,b)=(a,c)=(b,c)=1\\ b\mid u,\ a\mid u+j}} &A(a)A(b)A(c)K(u/b)K((u+j)/a)\\[-2pt] &\quad\cdot K\!\left(\frac{c(u/b)-1}{a}\right) \rho_0(\log c/B)\Psi_s(a/(Tc),b/(Tc)). \end{split} \tag{46}\] All coefficient variables are positive integers. Divisibility in \(\widehat{\mathbb Z}\) means membership in the image of multiplication by the divisor; that multiplication is injective. The preceding congruence calculation also proves that every quotient in (46) is defined on its summation domain. This kernel is nonnegative, depends on finitely many prime-power residues, and satisfies \[\mathcal K_j(u,s)=\mathcal K_{-j}(u+j,s).\] For this last identity, exchange \(a,b\): the last quotient is unchanged because \(c(u+j)-a=cu-b\).

Blocks and the norm used for comparison

Fix \(0<\eta<1\) and \(C_6>1\), and set \(M=\lceil C_6T\rceil\). Removing the lags \(|j|<\eta T\) in (45) costs \(O(\eta)+o_x(1)\) in absolute upper limit, by Lemma 22 and the bounded ordinary averages of \(\Sigma\). Average the origins over \(i=1,\ldots,M\), writing \(n'=n+i\). For fixed \(B\), replacing \((n+i)/(Tx)\) by \(n/(Tx)\) has vanishing error. At lag \(j\), the fraction of \(i\) for which \(k=i+j\notin[1,M]\) is at most \(|j|/M\). The total cost of these endpoints is \(O(T/M)=O(1/C_6)\). The remaining expression is \[ \mathcal Q^{\mathrm{arith}}_{B,x}:=\frac1{Tx}\sum_n\frac1M \sum_{\substack{1\le i,k\le M\\\eta T\le|k-i|\le T}} \overline{F_x(n+i)}F_x(n+k) \mathcal K_{k-i}(n+i,n/(Tx)). \tag{48}\] Thus (44) and the diagonal estimate give a positive lower bound for the iterated lower limit of (48), once \(\eta\) is small enough and \(C_6\) is large enough. These choices are made after \(C_*\) and before letting \(B\) grow.

Call a pair \((i,k)\) allowed if \(1\le i,k\le M\) and \(\eta T\le|k-i|\le T\). All matrices below are zero on other pairs. For a real matrix \(H\), define \[ \|H\|_\square= \frac1M\max_{\epsilon,\delta\in\{-1,1\}^M} \left|\sum_{i,k}H_{ik}\epsilon_i\delta_k\right|. \tag{49}\] The same maximum results from allowing real test coordinates in \([-1,1]\). Splitting real and imaginary parts shows that for complex \(z_i\) with \(|z_i|\le2\), \[ \left|\frac1M\sum_{i,k}H_{ik}\overline{z_i}z_k\right| \le16\|H\|_\square. \tag{50}\] For fixed \(B\), a matrix whose entries are finite-residue functions continuous in \(s\) has the same property after taking this finite maximum. Its ordinary scale averages converge to its Haar-product integral. Therefore bounds on the Haar expectation of the norm, uniformly for \(s\in\mathcal S\), control errors in (48). This comparison is deterministic in the labels; it requires no independence between the labels and the residue data.

Endpoint types and candidate representations

The block energy is now expressed in the norm that will control its approximation. We next separate the prime-divisibility data at the endpoints from the additional residue data belonging to each edge representation.

Put \(S_i(u)=\{p\in\mathcal P:p\mid u+i\}\). We compare them with independent sets \(S_i\), each formed by independently including every \(p\in\mathcal P\) with probability \(1/p\). At a fixed \(p>P_0>M\), the actual law has disjoint hits of probability \(1/p\) at each site. Its total variation distance from independent hits is \(O(M^2/p^2)\). Consequently the joint site laws can be coupled with failure probability \(O(M^2/P_0)\). Also \[ \mathbb P\{p^2\mid u+i\text{ for some }p\in\mathcal P,\ i\le M\} \ll M/P_0. \tag{51}\]

For an allowed pair \(i,k=i+j\) with \(j>0\), a candidate consists of a subset product \(b\) of \(S_i\), a subset product \(a\) of \(S_k\), and the positive integer \(c=(a-b)/j\). Require pairwise coprimality, rough squarefreeness, all three coefficient regularity conditions, and regularity of the two remaining sets \(S_i\setminus b\) and \(S_k\setminus a\). Here \(S\setminus b\) means removal of the primes dividing \(b\). It is harmless to use the closed support windows \[B\le\log c\le2B,\qquad Tc\le a,b\le2Tc;\] the smooth factors still impose the original supports. A reversed ordered pair refers to the same candidate.

A contributing endpoint has at most \((1+2\tau)\ell\) primes. There are at most \(2^{2(1+2\tau)\ell}\) subset choices at a pair, so the number of candidates in the entire block is \(O(B^4)\). This rough bound also applies to representations that contribute to (46) in the presence of site squares: the union of the coefficient and quotient prime sets covers the site’s distinct primes, and their total-count restrictions give the same bound. Together with (32), this gives, for example, a deterministic \(O(B^{10})\) bound for all total matrix masses used below. This bound allows small-probability errors to be discarded before sharper mean estimates are available.

Set \(k_B=\mathbb E_S K(S)\) under the independent \(1/p\) law. The untruncated Euler product gives \(k_B\ll1\). Define the latent kernel \[ \begin{split} \mathcal L_{ik}(S_i,S_k;s)=\frac{k_B}{B} \sum_{\text{candidates}}&A(a)A(b)A(c) K(S_i\setminus b)K(S_k\setminus a)\\[-2pt] &\quad\cdot\rho_0(\log c/B)\Psi_s(a/(Tc),b/(Tc)), \end{split} \tag{52}\] and extend it by symmetry and by zero on nonpairs. For actual endpoint sets without site squares, the terms allowed by the smooth factors have the same coefficient representations in (46) and (52), and the two endpoint quotient prime sets are the complementary sets displayed in (52). The latter kernel keeps these terms and replaces the remaining factor \(K(m)\) by its independent mean \(k_B\).

Separating the auxiliary roots

The following comparison justifies this averaging in expected cut norm, uniformly against a proposed endpoint kernel. It is stronger than an estimate against any one predetermined pair of tests.

The measure obtained from a first representation

Fix an allowed pair \(i,k=i+j\). Let \(\mathcal C_j\) be the finite set of positive triples \((a,b,c)\) satisfying \[a-b=jc,\qquad B\le\log c\le2B,\qquad Tc\le a,b\le2Tc,\] whose entries are pairwise coprime, rough, squarefree, and regular. These are precisely the coefficient conditions for a candidate; the two remainder regularity conditions still depend on the site types. We initially take \(j>0\); the negative case follows by exchanging the endpoints. Fix \((a,b,c)\in\mathcal C_j\). For an integer \(v\) whose prime factors belong to \(\mathcal P\), write \(\mathcal P(v)\) for its set of prime factors.

In the independent-site model, the event \(\mathcal P(b)\subset S_i\), \(\mathcal P(a)\subset S_k\) has probability \(1/(ab)\). Conditional on that event, put \[\mathcal T_b=S_i\setminus\mathcal P(b), \qquad \mathcal T_a=S_k\setminus\mathcal P(a).\] These sets are independent, and at each prime not excluded by the corresponding coefficient the inclusion probability is \(1/p\). Weighting one such inclusion by its factor \(1/2\) in \(K_0\) changes this probability to \[\frac{(1/p)/2}{1-1/(2p)}=\frac1{2p-1}.\] Thus weighting by \(K_0(\mathcal T_b)K_0(\mathcal T_a)\) gives a product probability measure, denoted by \(\mathbb P^{\mathrm{tilt}}_{a,b}\), under which the two remainder sets are independent and their available prime indicators have probabilities \(1/(2p-1)\).

For a coefficient \(v\) the normalizing factor is \[\Xi(v)=R^{1/2} \prod_{p\in\mathcal P\setminus\mathcal P(v)} \left(1-\frac1{2p}\right).\] The full product with no exclusions is bounded by Mertens’ estimate. Furthermore \(\log v=O(B)\) and every prime of \(v\) exceeds \(P_0\), so \[\sum_{p\mid v}\frac1p\ll\frac{B}{P_0}.\] Removing these factors changes the product by \(1+o(1)\). Consequently \(\Xi(v)\ll1\) uniformly for the coefficients under consideration. For every nonnegative function \(H\) of the two remainder sets we therefore have the exact change-of-measure identity \[\begin{split} &\mathbb E\big[ \mathbf 1_{\mathcal P(b)\subset S_i,\,\mathcal P(a)\subset S_k} K_0(\mathcal T_b)K_0(\mathcal T_a)H(\mathcal T_b,\mathcal T_a) \big]\\ &\hspace{25mm} =\frac{\Xi(b)\Xi(a)}{ab} \mathbb E^{\mathrm{tilt}}_{a,b}H(\mathcal T_b,\mathcal T_a). \end{split}\] The indicators that the two first remainders are regular will stay inside \(H\). In particular, we never divide by their probability.

Given these remainder sets, reconstruct \(S_i=\mathcal P(b)\cup\mathcal T_b\) and \(S_k=\mathcal P(a)\cup\mathcal T_a\), and let \(N_{a,b,c}(\mathcal T_b,\mathcal T_a)\) be the number of valid candidates at this pair for the reconstructed types, including the first triple whenever it is valid.

We can now see exactly how this measure enters the square. Expand the square of the sum of nonnegative candidate contributions, designate one candidate as the first representation, and bound every alternative contribution by (58). The first contribution retains the two factors \(K_0(\mathcal T_b)K_0(\mathcal T_a)\) and their regularity indicators. The change-of-measure identity gives \[ \begin{split} \mathbb E\,\mathcal L_{ik}^{2} &\ll B^{-\kappa_1+5\tau\log2+o(1)} \frac1B\sum_{(a,b,c)\in\mathcal C_j} \frac{A(a)A(b)A(c)}{ab}\,\Xi(a)\Xi(b)\\ &\qquad\cdot \mathbb E^{\mathrm{tilt}}_{a,b} \big[ \mathbf 1_{\mathcal T_b\ \mathrm{regular}} \mathbf 1_{\mathcal T_a\ \mathrm{regular}} N_{a,b,c}(\mathcal T_b,\mathcal T_a) \big]. \end{split} \tag{59}\] The implied constant absorbs \(k_B\) and the bounded smooth factors. Thus the remaining multiplicity question is the following uniform estimate; the coefficient sum will then be bounded by (31).

Squaring the kernel and averaging the rows

The multiplicity estimate has controlled how many alternative representations survive after weighting a first one. Combining it with the small weight of each representation now gives the required integrated row-square bound.

The channel and its two-split kernel

Let \(\Omega_B\) be the finite set of subsets of \(\mathcal P\), with probability measure under which the inclusions \(p\in S\) are independent and have probabilities \(1/p\). Given \(S\), retain each of its primes independently with probability \(1/2\), and let \(b\) be their product. Unconditionally \(b\) has law \(\nu_{1/2}\). Write \[I=[1/2,3],\qquad x_b=\frac{\log b}{B}.\] For a positive integer \(m_1\), partition \(I\) into \(m_1\) equal coarse intervals \(I_l\). Choose the number \(n_B\) of fine intervals to be \[n_B=m_1\left\lceil\frac{|I|B^{11/10}}{m_1}\right\rceil, \qquad \delta_B=\frac{|I|}{n_B}.\] Thus the fine partition refines the coarse partition and \(\delta_B\asymp B^{-11/10}\) for every fixed \(m_1\). Use consistent half-open conventions, assigning the final endpoint to the last cell.

For an integer \(1\le d\le B\), let \(\mathcal R_d=(\mathbb Z/d\mathbb Z)^\times\) with uniform probability measure; for \(d=1\) this is the one-point group. All products in question are units modulo \(d\), since \(P_0>B\) for sufficiently large \(B\). Give \(I\times\mathcal R_d\) the product measure \[d\lambda_d(x,r)=dx\,d{\rm unif}_{\mathcal R_d}(r).\] For \(g\in L^2(\Omega_B)\), define a function constant on each fine logarithmic cell by \[ U_dg(x,r)=\frac{\varphi(d)}{\delta_B} \mathbb E\!\left[g(S)\mathbf 1_{\{x_b\in I_x^{\rm fine},\ b\equiv r\pmod d\}}\right], \qquad x\in I_x^{\rm fine}\subset I . \tag{64}\] The expectation in this definition includes both the site and its fair split. The domain norm of \(U_d\) is the site \(L^2\) norm, and its range norm is that of \(L^2(\lambda_d)\).

Let \(P_{\rm av}\) average the residue coordinate. Then \[u_g:=P_{\rm av}U_dg=U_1g\] is independent of \(d\). Let \(P_{m_1}\) average on the coarse logarithmic intervals and act identically on the residue coordinate when one is present. Let \(P_{\rm fine}\) denote averaging on the fine logarithmic intervals. These averaging operators are orthogonal projections, and \(P_{\rm fine}\) commutes with \(P_{m_1}\).

We first identify the kernel used to prove the proposition. A cell \(A=J\times\{r\}\), with \(J\) a fine logarithmic interval, has measure \[\lambda_d(A)=\frac{\delta_B}{\varphi(d)}=:\mu_d.\] Set \[p_A(S)=\mathbb P_{\rm split}(x_b\in J,\ b\equiv r\pmod d\mid S).\] For a step function \(F\) in the range space, the adjoint of the channel is \[U_d^*F(S)=\sum_A p_A(S)F|_A.\] Consequently \(U_dU_d^*\) has the step kernel \[ K_d(A,A')=\frac{\mathbb E[p_A(S)p_{A'}(S)]}{\mu_d^2}. \tag{67}\] This is the density on fine cells of two conditionally independent fair splits of the same site, restricted to \(I\) in each logarithmic coordinate. The kernel is nonnegative and symmetric. Its row integral satisfies \[\sum_{A'}K_d(A,A')\lambda_d(A') =\frac{\mathbb E[p_A(S)\sum_{A'}p_{A'}(S)]}{\mu_d} \le\frac{\mathbb P_{\nu_{1/2}}(x_b\in J,\ b\equiv r\pmod d)}{\mu_d}.\] By (26), the numerator is at most \(C\delta_B/\varphi(d)+O(B^{-80})\). Indeed its logarithmic interval is contained in the fixed compact interval \(I\), where the densities are uniformly bounded. Since \(\mu_d^{-1}\ll B^{21/10}\), the row integrals are bounded uniformly. The column integrals have the same bound. Schur’s inequality gives \(\|U_dU_d^*\|\ll1\), and therefore proves the first assertion of (65). This proof already applies to every \(L^2\) input, including signed or complex inputs.

Removing small exclusive products

The two-split model reduces the channel operator to independent common and exclusive products. An exclusive product near \(1\) provides too little averaging to erase its residue, so we first bound the operator contribution of those products. Both row and column bounds are needed.

For a product \(v\), write \(\ell_B(v)=(\log v)/B\). Let \(K_d^{\rm ind}\) be the independent-product kernel from Lemma 28. Its entries are \[K_d^{\rm ind}(A,A') =\frac{\mathbb P((\ell_B(CE_1),CE_1\bmod d)\in A,\ (\ell_B(CE_2),CE_2\bmod d)\in A')}{\mu_d^2}.\] The following estimate concerns the operator obtained by retaining only the specified event in this probability.

The separate column estimate is essential: an estimate for the total probability of a discarded event would not by itself control its operator norm.

Flattening all nonconstant residue modes

We prove the second assertion of (65). Set \(\theta=B^{-1/10}\) and retain the part of \(K_d^{\rm ind}\) where \(\ell_B(E_1),\ell_B(E_2)>\theta\). For a fixed common product \(C=c\), the probability that one retained exclusive lands in the output cell \(A=J\times\{r\}\) is \[p_A^{(\theta)}(c) =\mathbb P\bigl(\ell_B(E)\in(J-\ell_B(c))\cap(\theta,\infty),\ E\equiv rc^{-1}\pmod d\bigr).\] If the interval is nonempty it is contained in \([B^{-1/10},3]\), since \(\ell_B(c)\ge0\). The interval and its endpoint conventions, including a partial cell at \(\theta\), are within the uniform statement of (26). Consequently \[ p_A^{(\theta)}(c) =\frac1{\varphi(d)} \int_{(J-\ell_B(c))\cap(\theta,\infty)}f_{1/4,B}(t)\,dt +O(B^{-80}). \tag{68}\] The main term depends on \(J\) and \(\ell_B(c)\), but not on \(r\) or on the residue of \(c\). All terms are uniformly bounded: the probability is at most one and its main term differs by \(O(B^{-80})\). Alternatively the bound \(f_{1/4,B}(t)\ll t^{-3/4}\) gives the explicit cell bound \(O(\delta_B B^{3/40}/\varphi(d))\).

Conditional on \(C\), the exclusives are independent. Multiplying (68) for two output cells and integrating over \(C\), the retained kernel differs from a kernel independent of both residues by at most \[O(B^{-80}\mu_d^{-2})=O(B^{-75})\] pointwise, and hence by \(O(B^{-70})\) in operator norm. A kernel independent of both residues is annihilated by projection onto \(1-P_{\rm av}\) on either side. Lemmas 28 and 29 therefore yield \[\begin{split} \|(1-P_{\rm av})U_dU_d^*(1-P_{\rm av})\| &\ll (B^{-1/10}+1/R)^{1/4} +B^{-70}+B^5/P_0\\ &\ll B^{-1/40}, \end{split}\] because \(1/R=(\log P_0)/B=O(\log B/B)\) and \(P_0=B^{1000}\). Taking the square root of this operator identity gives the stronger bound \[\|(1-P_{\rm av})U_d\|\ll B^{-1/80}.\] In particular (65) holds with \(c_7=1/200\). This reasoning did not fix an input function at any stage, so the estimate is uniform on the whole unit ball of \(L^2(\Omega_B)\).

Uniform coarse compactness

Residue dependence is now negligible uniformly in the input function. The remaining task is to approximate the logarithmic output uniformly on a fixed finite partition. This will give a finite family of endpoint features for the graph kernel.

We now prove (66). It suffices to use \(d=1\), since \(u_g=U_1g\). Fix \(0<\theta<1/20\) and choose a smooth function \(a_\theta:\mathbb R\to[0,1]\) equal to zero on \((-\infty,\theta]\) and to one on \([2\theta,\infty)\). It may be chosen nondecreasing. In the independent-product kernel insert the weight \(a_\theta(\ell_B(E_1))a_\theta(\ell_B(E_2))\). The change is a nonnegative kernel supported where at least one exclusive logarithm is at most \(2\theta\). Lemma 29 bounds its operator norm by \[O((2\theta+1/R)^{1/4})+O(B^{-70}+B^5/P_0).\]

Choose also a smooth upper cutoff equal to one on \([0,3]\) and supported below \(16/5\). On the positive axis let \(f_{1/4,B}^{\rm cut}\) be the product of this upper cutoff, \(a_\theta\), and \(f_{1/4,B}\), extended by zero to the real line. For each fixed \(\theta\), all derivatives of any fixed order of this function are bounded uniformly in \(B\). This follows from the derivative bounds for \(f_{1/4,B}\) established before Proposition 11; the cutoff keeps the argument away from zero. The upper cutoff changes none of the probabilities relevant to the output interval \(I\).

For completeness, the weighted version of the local law used here follows directly from its interval version. For fixed \(z=\ell_B(c)\) and a fine interval \(J\), apply (26) to subintervals of \((J-z)\cap[\theta,3]\). The difference between the exclusive-product measure and the density \(f_{1/4,B}(t)\,dt\) has cumulative integral \(O(B^{-80})\) there. Stieltjes integration by parts against \(a_\theta\) bounds its weighted integral by \(O_\theta(B^{-80})\): the endpoint values and the total variation of \(a_\theta\) are bounded. Thus, uniformly in \(c,J\), \[\mathbb E\!\left[ a_\theta(\ell_B(E))\mathbf 1_{\{\ell_B(cE)\in J\}}\right] =\int_J f_{1/4,B}^{\rm cut}(x-\ell_B(c))\,dx +O_\theta(B^{-80}).\] Conditional independence of the two exclusives now identifies the weighted step kernel, up to \(O_\theta(B^{-70})\) in operator norm, with the fine-cell compression of the continuous kernel \[ K_{B,\theta}(x,y)= \mathbb E_C\!\left[ f_{1/4,B}^{\rm cut}(x-\ell_B(C)) f_{1/4,B}^{\rm cut}(y-\ell_B(C))\right], \qquad (x,y)\in I^2. \tag{69}\] Indeed multiplying the two weighted cell formulas makes an \(O_\theta(B^{-80})\) error in each cell-pair probability; division by \(\delta_B^2\) still leaves an operator error smaller than \(O_\theta(B^{-70})\).

The kernel in (69) and its first derivatives are bounded uniformly in \(B\), with constants depending on \(\theta\). Differentiation may be taken under the expectation because the cut densities and their derivatives have uniform bounds and the law of \(C\) is a probability measure. In particular no limiting law for \(C\), and no derivative bound for that law, is needed. Let \(\mathcal T_{B,\theta}\) be the integral operator with this kernel. If \(a=|I|/m_1\) is the coarse mesh, the mean value theorem gives \[\sup_{x,y\in I} \left|K_{B,\theta}(x,y) -(P_{m_1}^{(x)}P_{m_1}^{(y)}K_{B,\theta})(x,y)\right| \ll_\theta a.\] Here the two superscripts denote averaging the indicated kernel coordinate. Schur’s inequality consequently gives \[ \|\mathcal T_{B,\theta} -P_{m_1}\mathcal T_{B,\theta}P_{m_1}\| \ll_\theta m_1^{-1}. \tag{70}\]

Write \(Q=1-P_{m_1}\). Fine and coarse averaging commute, and fine averaging is a contraction. Therefore \[\|Q P_{\rm fine}\mathcal T_{B,\theta}P_{\rm fine}Q\| \le \|Q\mathcal T_{B,\theta}Q\| \ll_\theta m_1^{-1}.\] Combining this with the independent-model comparison and the cutoff estimate proves, for every fixed \(\theta,m_1\), \[\begin{split} \|Q U_1\|^2 &=\|Q U_1U_1^*Q\|\\ &\le C(2\theta+1/R)^{1/4} +C_\theta m_1^{-1}+o_{\theta,m_1}(1). \end{split}\] Given \(\epsilon_1>0\), first choose \(\theta\) so that \(C(2\theta)^{1/4}<\epsilon_1^2/4\), and then choose \(m_1\) so that \(C_\theta/m_1<\epsilon_1^2/4\). Taking the upper limit in \(B\) and the square root proves (66), and completes the proof of Proposition 27.

Removing the regularity restrictions

Let \(S\in\Omega_B\) have the independent-site law: each prime in \(\mathcal P\) belongs to \(S\) independently with probability \(1/p\). A fair split of \(S\) selects each of its primes independently with probability \(1/2\); write \(b\) for the product of the selected primes. The unconditional law of \(b\) is \(\nu_{1/2}\). For a real measurable function \(g\) on the site space, define the signed mass \[\gamma_g(n)=\mathbb E\big[g(S)\mathbf 1_{\{b=n\}}\big].\] In particular, \(\lvert\gamma_g(n)\rvert\le\nu_{1/2}(n)\) whenever \(\lvert g\rvert\le1\). All statements below allow \(g\) to depend on \(B\).

For a subset product \(b\) of \(S\), the untruncated weights satisfy the exact identity \[A_0(b)K_0(S\setminus b) = (\log P_0)R\,2^{-|S|}=B\,2^{-|S|}.\] The factor \(2^{-|S|}\) is the conditional probability of each fair split. This identity connects the endpoint weights with the unrestricted splits defining the channels. The next lemma bounds the cost of removing the regularity indicators inside the integrated test; after that removal, the endpoint averages are the unrestricted ones defining the channels.

Fourier detection and the discarded arcs

The support of (72) already predicts the structure of its real-variable comparison. For a nonzero term put \[x=\frac{\log a}{B},\qquad y=\frac{\log b}{B},\qquad u_1=\frac{a}{Tc},\qquad u_2=\frac{b}{Tc}.\] The exact relation \(a-b=jc\) gives \[u_1-u_2=\frac jT,\qquad B(x-y)=\log(u_1/u_2).\] Since \(u_1,u_2\) lie in a fixed compact subinterval of \((1,2)\), the two log coordinates satisfy \(0<x-y\ll B^{-1}\). We will enforce the discrete relation by additive Fourier orthogonality. This places the \(c\) sum in a factor to which the arithmetic estimates apply, while the two signed endpoint sums are controlled by their mass and \(L^2\) bounds. For the surviving small denominators, the channel estimates allow us to average the residue dependence of the endpoint tests with a uniform error, leaving an explicit finite residue sum. Fourier inversion will then give an integral on the narrow logarithmic band, where the coarse channel approximation produces the feature products described above.

We first analyze (72) in a single smooth dyadic box. Choose a smooth compactly supported dyadic partition of unity on \((0,\infty)\) for the variable \(c\), and write \(X=2^m\), \(N_X=TX\). A box has \(c/X\) in a fixed compact subinterval of \((0,\infty)\). On the support of \(\Psi_s\), both \(a/N_X\) and \(b/N_X\) are also in fixed positive compact intervals. The smooth weight in the three variables \[(a/N_X,b/N_X,c/X)\] has all fixed-order derivatives bounded uniformly in \(m,B,s\). This includes \(\rho_0((\log X+\log(c/X))/B)\), whose differentiated log-scale factors only improve the bound.

Extend this weight smoothly inside a larger fixed cube, expand the extension in a periodic Fourier series, and multiply each coordinate factor by a fixed smooth compact cutoff equal to one on the original support. We obtain a sum of separated products \[w_1(a/N_X)w_2(b/N_X)w_3(c/X).\] More precisely, if \(\boldsymbol\nu\in\mathbb Z^3\) indexes the Fourier terms, their scalar coefficients are \(O_A((1+|\boldsymbol\nu|)^{-A})\) for every fixed \(A\), uniformly in the boxes and in \(s\). The fixed-order smooth norms of the factors grow at most polynomially in \(|\boldsymbol\nu|\). All estimates below have only finitely many such smooth-norm losses; choosing \(A\) larger than those losses plus four makes every Fourier sum absolutely convergent. When a bound is summed over boxes, we use the uniform coefficient bound at each fixed \(\boldsymbol\nu\). It therefore suffices to write the calculation for one separated term.

For that term set \[\begin{split} S_1(\alpha)&=\sum_a\gamma_h(a)w_1(a/N_X)e(a\alpha),\\ S_2(\alpha)&=\sum_b\gamma_g(b)w_2(b/N_X)e(b\alpha),\\ S_3(\alpha)&=\sum_c A_0(c)w_3(c/X)e(c\alpha). \end{split}\] Additive orthogonality expresses its contribution as \[k_BB\int_0^1 S_1(\alpha)S_2(-\alpha)S_3(-j\alpha)\,d\alpha.\] The elementary product formula and Mertens’ estimate give \[0\le k_B\le\mathbb EK_0(S) =R^{1/2}\prod_{p\in\mathcal P}(1-1/(2p))\ll1,\] uniformly in the truncation constant. By (28), (29), and Parseval, \[ \int_0^1|S_i(\alpha)|^2\,d\alpha \ll\frac1{B^2N_X^2}\sum_{n\asymp N_X}A_0(n)^2 \ll\frac{B^{-2+1/4+o(1)}}{N_X},\qquad i=1,2. \tag{73}\] The first-moment part of (29) also gives \[ \sup_\alpha|S_i(\alpha)|\ll B^{-1},\qquad i=1,2. \tag{74}\] The implicit constants here and below include the specified smooth norms of the separated factors.

On the circle of \(j\alpha\pmod1\), take major arcs of radius \(B^{13}/X\) about the reduced fractions \(u/q\) with \(q\le B^{12}\). They are disjoint for large \(B\): the distance between distinct such fractions is at least \(B^{-24}\), whereas \(X\) is exponential in \(B\). On the complement we have \[ |S_3(-j\alpha)|\ll X B^{-1/2+o(1)}. \tag{75}\] Here are the details of the imported estimate and its application. The Montgomery–Vaughan bound (Montgomery and Vaughan 1977, Corollary 1) states that a multiplicative function \(f\) satisfying \(\lvert f(p)\rvert\le1\) for every prime \(p\) and \(\sum_{n\le Z}|f(n)|^2\le Z\) for every \(Z\ge1\) obeys \[\left|\sum_{n\le Y}f(n)e(n\theta)\right| \ll \frac{Y}{\log Y} +\frac{Y(\log R')^{3/2}}{\sqrt{R'}}\] whenever \(\lvert\theta-u'/q'\rvert\le1/(q')^2\), \((u',q')=1\), and \(2\le R'\le q'\le Y/R'\). Apply Dirichlet approximation with denominator limit \(\lfloor X/B^{12}\rfloor\). If the resulting denominator is at most \(B^{12}\), the approximation error is at most \(2B^{12}/X<B^{13}/X\), so the point is on a major arc. Off the major arcs we consequently have \[B^{12}<q'\le X/B^{12},\qquad |\theta-u'/q'|\le1/(q')^2.\] For every \(Y\asymp X\) arising in partial summation, these inequalities permit \(R'=B^{10}\). The function \(f(n)=\mu_{\rm Mob}^2(n)2^{-\omega(n)}\mathbf 1_{\{n\ {\rm rough}\}}\) is multiplicative and 1-bounded, so it satisfies the theorem’s hypotheses. Smooth partial summation and restoration of the factor \[(\log P_0)R^{1/2}=\sqrt{B\log P_0}=B^{1/2+o(1)}\] give (75).

For later reference, if a discarded region satisfies \(\lvert S_3\rvert\ll XB^{-a+o(1)}\), its contribution in one box is at most \[ C B\,XB^{-a+o(1)} \frac{B^{-7/4+o(1)}}{TX} =\frac{B^{-3/4-a+o(1)}}{T}, \tag{76}\] by Cauchy–Schwarz and (73). Summing over the \(O(B)\) boxes gives a total minor-arc error of \[\frac{B^{-1/4+o(1)}}{T}.\]

On a major arc, apply (25) to \(S_3\). When \(q>B^{0.4}\), its main term is at most \(X/\varphi(q)\ll XB^{-0.4+o(1)}\), uniformly along the arc. Formula (76) shows that these arcs cost \(B^{-0.15+o(1)}/T\) in total. The error \(O(XB^{-50})\) in (25), on all the major arcs together, costs at most \(B^{-49.75+o(1)}/T\). All three errors are uniform in the endpoint tests and are \(o(1/T)\).

Put \(Q=B^{0.4}\). The remaining arcs for \(\alpha\) have the following exact parametrization: \[d=jq,\qquad h'\pmod d,\qquad (h',q)=1,\qquad \alpha=\frac{h'}d+\frac{\xi}{jX},\qquad |\xi|\le B^{13}.\] For each reduced fraction on the \(j\alpha\) circle there are exactly \(j\) lifts in this list. In particular the parametrization neither omits nor repeats an arc. Define \[\widehat w_{3,B,X}(\xi) =\int w_3(z)m_B(\log(Xz)/B)e(-\xi z)\,dz.\] These transforms are uniformly Schwartz, with bounds controlled by fixed smooth norms. The main term from (25) contains the factor \(X\), while \(d\alpha=d\xi/(jX)\). The resulting box expression is therefore \[ \frac{k_BB}{j} \sum_{q\le Q}\frac{\mu_{\rm Mob}(q)}{\varphi(q)} \sum_{\substack{h'\bmod d\\(h',q)=1}} \int_\mathbb R\widehat w_{3,B,X}(\xi) S_1\left(\frac{h'}d+\frac{\xi}{jX}\right) S_2\left(-\frac{h'}d-\frac{\xi}{jX}\right)\,d\xi. \tag{77}\] In this formula the integration has been extended from \([-B^{13},B^{13}]\) to \(\mathbb R\). To justify it, use (74), the bound of \(j\varphi(q)\) for the number of lifts at denominator \(q\), and a Schwartz bound of any sufficiently large fixed order. The number of boxes and the total arc count are polynomial in \(B\), so the tails give \(o(1/T)\) after all sums. Notice also that \(d\le TQ\le B^{0.72}<B\), as required for the channel estimates.

Fine histograms and residue averaging

The discarded arcs already have negligible total contribution. The remaining major arcs carry a logarithmic coordinate and a unit residue at each endpoint. We next use the channel bounds to retain the former while averaging the latter, uniformly for the endpoint tests.

We replace the split-product measures in (77) by the fine log and residue histograms of (64). For instance, the replacement for its first sum is \[ \frac1{\varphi(d)}\sum_{r\in\mathcal R_d}e(h'r/d) \int_I U_dh(x,r)w_1(e^{Bx}/N_X) e(\xi e^{Bx}/(jX))\,dx. \tag{78}\] The residue phases have not been approximated. We give the norm estimate which controls this step, its denominator sum, and the subsequent residue projection.

Let \(I_X\subset I\) be an enlarged log window about \(\log N_X/B\), of length \(O(1/B)\), which contains all fine cells meeting the support of the factors in a box. These windows have bounded overlap as \(X\) ranges over its dyadic values, because successive centers are spaced by \((\log2)/B\). They lie in the interior of \(I\) for large \(B\). If \(F\) is supported in \(I_X\) and bounded, finite Fourier Parseval on \(\mathbb Z/d\mathbb Z\), with the residue vector extended by zero off the units, gives \[ \begin{split} &\sum_{h'\bmod d}\left| \frac1{\varphi(d)}\sum_{r\in\mathcal R_d}e(h'r/d) \int_{I_X}V(x,r)F(x)\,dx\right|^2\\ &\hspace{12mm}\le \frac d{\varphi(d)}\,|I_X|\,\|F\|_\infty^2 \|V\|_{L^2(I_X\times\mathcal R_d)}^2. \end{split} \tag{79}\] Indeed the exact finite Fourier identity, before applying Cauchy–Schwarz to the integrals, is \[\sum_{h'\bmod d} \left|\frac1{\varphi(d)}\sum_{r\in\mathcal R_d}e(h'r/d)v_r\right|^2 =\frac d{\varphi(d)}\frac1{\varphi(d)} \sum_{r\in\mathcal R_d}|v_r|^2.\] Here and throughout, the residue component of the \(L^2\) norm uses uniform probability on \(\mathcal R_d\).

For clarity, the histogram replacement can be estimated even when the signed masses \(\gamma_g\) have no regularity whatsoever. For a fine cell \(J\) and a unit residue \(r\), the definition (64) says exactly that \[\frac1{\varphi(d)}\int_J U_dh(x,r)\,dx =\mathbb E\big[h(S)\mathbf 1_{\{\log b/B\in J,\ b\equiv r\ (d)\}}\big].\] For \(F(x)=w_1(e^{Bx}/N_X)e(\xi e^{Bx}/(jX))\), differentiation on the relevant enlarged window gives \[\|F'\|_\infty\le C_{w,\eta}(1+|\xi|)B.\] We used \(e^{Bx}\asymp TX\) and \(j\ge\eta T\). Thus the oscillation of \(F\) on a fine cell is at most \[a_B(\xi):=C_{w,\eta}(1+|\xi|)B\delta_B.\] After taking the factor \(1/\varphi(d)\) outside as in (78), the residue vector of the replacement error has absolute value at residue \(r\) at most \[a_B(\xi)\int_{I_X}U_d1(x,r)\,dx.\] This follows by subtracting the cell average of \(F\) from its value at the actual split product and using \(\lvert h\rvert\le1\). Positivity of the measure for \(U_d1\) is the only pointwise information used. Formula (79) therefore bounds the squared sum of this error over \(h'\) by \[C\frac d{\varphi(d)}|I_X|\,a_B(\xi)^2 \|U_d1\|_{L^2(I_X\times\mathcal R_d)}^2.\] The actual, unreplaced sum has the same type of estimate without \(a_B\), by bounding its residue vector with \(\|F\|_\infty\int_{I_X}U_d1\). The replaced sum has (79) directly with \(V=U_dh\).

We apply these estimates to the difference of the two products in (77), replacing one factor at a time. Cauchy–Schwarz over \(h'\) is valid even though the sum is restricted by \((h',q)=1\), since extending either squared sum to all residues increases it. Afterwards Cauchy–Schwarz over the boxes and bounded overlap of \(I_X\) bound the sum of products of local norms by the product of global norms. These global norms are bounded by (65). Finally the Schwartz moments absorb the factors \(1+|\xi|\), as well as the polynomial smooth losses in the separated expansion. Since \(|I_X|\ll1/B\), this proves that at a fixed \(q\) the total histogram error, including all boxes, is at most \[ \frac{C_\eta}{j} \frac1{\varphi(q)}\frac{jq}{\varphi(jq)}\,B\delta_B. \tag{80}\] In particular, the number \(O(B)\) of boxes has not introduced an extra factor of \(B\): the local window length cancels the \(B\) in (77), and the remaining local norms are summed by bounded overlap.

We record explicitly a bound for the denominator sum. The elementary inequality \[\frac{jq}{\varphi(jq)} \le\frac j{\varphi(j)}\frac q{\varphi(q)}\] and partial summation imply \[ \sum_{q\le Q}\frac1{\varphi(q)}\frac{jq}{\varphi(jq)} \ll\frac j{\varphi(j)}\log(2Q). \tag{81}\] For completeness, to verify the required mean-value input write \((n/\varphi(n))^2=\sum_{d\mid n}b(d)\), where \(b\) is supported on squarefree integers and \[b(p)=(1-1/p)^{-2}-1=O(1/p).\] All these coefficients are nonnegative, and \[\sum_d\frac{b(d)}d=\prod_p(1+b(p)/p)<\infty.\] Hence \(\sum_{q\le Q}(q/\varphi(q))^2\ll Q\), which after partial summation gives (81). Since \(j/\varphi(j)=B^{o(1)}\) uniformly for \(j\le T\) and \(B\delta_B\asymp B^{-1/10}\), summing (80) gives \(o(1/T)\), uniformly on the allowed lag range.

We now replace \(U_dh,U_dg\) in the histogram expression by their residue averages \(u_h,u_g\). Expand the difference of products one factor at a time and repeat (79). The only new input is \[\|U_dh-u_h\|_2+\|U_dg-u_g\|_2\ll B^{-c_7}\] from (65), uniformly in \(d\le B\) and the bounded tests. It follows that the error is bounded by \[\frac{C_\eta}{j}\frac j{\varphi(j)} \log(2Q)B^{-c_7}=o(1/T).\] This establishes the residue projection for arbitrary bounded single-site tests, rather than just for the constant test.

The exact residue factor

After the channel projection, the endpoint tests no longer depend on a residue coordinate. The remaining finite residue sum can therefore be computed exactly; it is the arithmetic factor governing the allowed lags.

After residue averaging, the unit phase average in (78) is \(c_d(h')/\varphi(d)\), where \[c_d(h')=\sum_{r\in\mathcal R_d}e(h'r/d)\] is the Ramanujan sum. The second endpoint contributes its complex conjugate phase average. Since the coefficient of a nonsquarefree \(q\) in (77) is zero, only squarefree \(q\) matter. For such \(q\) the exact identity is \[ \sum_{\substack{h'\bmod jq\\(h',q)=1}} \frac{|c_{jq}(h')|^2}{\varphi(jq)^2} =\begin{cases} \displaystyle\frac{j}{\varphi(j)\varphi(q)},&(j,q)=1,\\[4pt] 0,&(j,q)>1. \end{cases} \tag{82}\] Here is a direct proof that also covers nonsquarefree \(j\). If \(p\mid(j,q)\), the modulus \(jq\) is divisible by \(p^2\), while \((h',q)=1\) forces \(p\nmid h'\). For every \(a\ge2\), \[c_{p^a}(h') =\sum_{r\bmod p^a}e(h'r/p^a) -\sum_{r\bmod p^{a-1}}e(h'r/p^{a-1})=0 \quad(p\nmid h').\] Chinese remaindering therefore makes the whole Ramanujan sum zero. If \((j,q)=1\), the Ramanujan sums split over the two coprime moduli. At the squarefree modulus \(q\), a unit frequency has \(c_q(h')=\mu_{\rm Mob}(q)\), of absolute value one. At modulus \(j\), orthogonality gives \[\sum_{h'\bmod j}|c_j(h')|^2 =\sum_{r,r'\in\mathcal R_j} \sum_{h'\bmod j}e(h'(r-r')/j)=j\varphi(j).\] The numerator on the left of (82) is thus \(j\varphi(j)\varphi(q)\), and division by \(\varphi(jq)^2=\varphi(j)^2\varphi(q)^2\) proves the identity.

Combining (82) with the factor \(\mu_{\rm Mob}(q)/\varphi(q)\) in (77) yields the singular series \[ \mathfrak S(j) =\frac j{\varphi(j)} \sum_{(q,j)=1}\frac{\mu_{\rm Mob}(q)}{\varphi(q)^2} =\frac j{\varphi(j)} \prod_{p\nmid j}\left(1-\frac1{(p-1)^2}\right). \tag{83}\] The series is absolutely convergent: for every fixed \(\epsilon>0\), \(1/\varphi(q)\ll_\epsilon q^{-1+\epsilon}\), so \[\sum_{q>Q}\frac1{\varphi(q)^2} \ll_\epsilon Q^{-1+2\epsilon} \quad(0<\epsilon<1/2).\] This tail bound is independent of \(j\). Also \[0\le\mathfrak S(j)\le j/\varphi(j).\] The series vanishes when \(j\) is odd, consistently with the parity obstruction in \(a-b=jc\) for rough coefficients.

After the residue projection, the archimedean integrals in (77) no longer depend on \(q\). The total over boxes of their absolute values, before the factor \(B/j\), is \(O(1/B)\): apply Cauchy–Schwarz on their log windows, then use bounded overlap and \(\|u_g\|_2,\|u_h\|_2\ll1\). The Schwartz integrations and smooth expansions have the same summability as before. It follows that replacing the sum \(q\le Q\) by (83) costs at most \[ \frac Cj\frac j{\varphi(j)} \sum_{q>Q}\frac1{\varphi(q)^2}. \tag{84}\] In particular this is a uniform vanishing multiple of \((j/\varphi(j))/j\).

Archimedean inversion and coarse features

For one separated box, the phase remaining in the product of archimedean integrals is \[e\left(\xi\frac{e^{Bx}-e^{By}}{jX}\right).\] Fourier inversion in \(\xi\) therefore evaluates \(w_3(z')m_B(\log(Xz')/B)\) at \(z'=(e^{Bx}-e^{By})/(jX)\). The inversion introduces no further Jacobian: its integration variable is \(\xi\). Recombining the separated factors and the dyadic partition gives \[ \begin{split} &\frac{k_BB}{j}\mathfrak S(j) \int_{\substack{x>y\\x,y\in I}}u_h(x)u_g(y)m_B(z)\rho_0(z)\\ &\hspace{14mm}\cdot \Psi_s\left( \frac{\lambda e^{B(x-y)}}{e^{B(x-y)}-1}, \frac{\lambda}{e^{B(x-y)}-1}\right)\,dx\,dy,\\ &\hspace{8mm}\lambda=\frac jT,\qquad z=\frac1B\log\left(\frac{e^{Bx}-e^{By}}j\right). \end{split} \tag{85}\] Weights are interpreted as zero off their positive domains and supports. The inversions and rearrangements are justified for \(u_g,u_h\in L^2(I)\) by the bounded log windows, the Schwartz transforms, and the absolutely summable smooth expansion, with the absolute-product bound just used for (84).

We spell out the support properties needed to approximate (85). There is a compact interval \([a_0,b_0]\subset(1,2)\) containing the support of \(\psi\). Write \(v=B(x-y)\). If the \(\Psi_s\) factor is nonzero, its two arguments \(u_1,u_2\) lie in \([a_0,b_0]\), with \[u_1-u_2=\lambda,\qquad e^v=u_1/u_2.\] For \(\eta\le\lambda\le1\), this implies \[ 0<c_\eta\le v\le C, \qquad c_\eta=\log(1+\eta/b_0),\quad C=\log(b_0/a_0). \tag{86}\] If these inequalities have no solution for a particular \(\lambda\), the weight is simply zero. On this support, \[z=y+\frac{\log(e^v-1)-\log j}{B} =y+O_\eta(\log T/B).\] The functions \(m_B\rho_0\), extended by zero beyond a fixed compact interval inside \((0,4)\), converge uniformly to \(m\rho_0\), with uniformly bounded derivatives. It follows that \[m_B(z)\rho_0(z)=m(y)\rho_0(y)+o_\eta(1)\] uniformly wherever the integrand can be nonzero.

To quantify its effect for arbitrary tests, consider the integral operator with nonnegative kernel \[B\mathbf 1_{\{0<x-y<C/B\}},\qquad x,y\in I.\] Its row and column integrals are at most \(C\), so Schur’s inequality gives a uniform \(L^2\) operator bound. Multiplication by the bounded smooth weights preserves that bound. Since \(\|u_g\|_2,\|u_h\|_2\ll1\), the replacement of \(m_B(z)\rho_0(z)\) in (85) costs \(o_\eta(1)\mathfrak S(j)/j\).

Given \(\epsilon_1>0\), choose the fixed coarse partition in (66), and write \(u_g^{\rm c}=P_{m_1}u_g\), \(u_h^{\rm c}=P_{m_1}u_h\). The same operator bound and the contraction property of \(P_{m_1}\) show that replacing the two endpoint functions by \(u_g^{\rm c},u_h^{\rm c}\) costs at most \[C_\eta(\epsilon_1+o(1))\frac{\mathfrak S(j)}j.\] This constant does not grow with the chosen coarse partition; the errors tending to zero may of course depend on that fixed partition. Its coarse functions satisfy \[\|u_g^{\rm c}\|_\infty,\ \|u_h^{\rm c}\|_\infty \le \max_l |I_l|^{-1/2} \max(\|u_g\|_2,\|u_h\|_2)\ll_{m_1}1.\] By (86), \(x\) and \(y\) belong to the same coarse cell except when \(y\) is within \(C/B\) of one of the finitely many cell boundaries. The area of these exceptional pairs is \(O_{m_1}(B^{-2})\): the total possible length for \(y\) is \(O_{m_1}(B^{-1})\), and for each such \(y\) the possible length for \(x\) is \(O(B^{-1})\). Thus replacing \(u_h^{\rm c}(x)\) by \(u_h^{\rm c}(y)\) costs \(O_{\eta,m_1}(B^{-1})\mathfrak S(j)/j\). The support of \(m(y)\rho_0(y)\) is a fixed positive distance from the boundary of \(I\), so there is no further boundary contribution from the condition \(x\in I\) for large \(B\).

Recall the coarse features from Corollary 30, and define their deterministic coefficients by \[V_l(S)=\frac1{|I_l|} \mathbb P_{\rm split}(\log b/B\in I_l\mid S), \qquad h_l=\int_{I_l}m(y)\rho_0(y)\,dy.\] These are nonnegative, \(V_l(S)\le |I_l|^{-1}\), and \(h_l\ge0\). Since the fine partition refines the coarse partition, averaging (64) over a coarse cell gives the exact identity \[(P_{m_1}u_g)|_{I_l}=\mathbb E[g(S)V_l(S)].\] After the preceding replacements in (85), substitute \(x=y+v/B\). The factor \(dx=dv/B\) cancels its prefactor \(B\), and \(1/j=(1/T)(1/\lambda)\). The resulting integral tests are therefore precisely those of the following site matrix: \[ \begin{split} \mathcal M_{ik}(S_i,S_k;s) &=\frac{k_B}{T}\mathfrak S(|j|) w(s,|j|/T)\sum_l h_lV_l(S_i)V_l(S_k),\qquad j=k-i,\\ w(s,\lambda) &=\frac1\lambda\int_{v>0} \Psi_s\left(\frac{\lambda e^v}{e^v-1}, \frac{\lambda}{e^v-1}\right)\,dv, \qquad \eta\le\lambda\le1. \end{split} \tag{87}\] The matrix is zero off the allowed pairs. Formula (87) defines it for both signs of \(j\); it is symmetric because its displayed dependence on the endpoints is symmetric. For positive \(j\), independence of the endpoint types turns its integral test against \(g,h\) into the product of the two expectations just displayed. The case of negative \(j\) follows by exchanging the endpoints.

The function \(w\) is nonnegative and smooth, with all fixed-order derivatives bounded on the relevant compact \(s\)-range and \(\eta\le\lambda\le1\). There is in fact an expression giving bounds independent of \(\eta\). In (87) put \(u=\lambda/(e^v-1)\). Then \(dv=-\lambda\,du/(u(u+\lambda))\), and hence \[ w(s,\lambda)=\int_0^\infty \frac{\Psi_s(u+\lambda,u)}{u(u+\lambda)}\,du. \tag{88}\] The factors \(\psi(u)\psi(u+\lambda)\) restrict the integration to a fixed compact subset of \(u>0\). Consequently this formula extends \(w\) smoothly to \(0\le\lambda\le1\), with uniformly bounded derivatives there, and gives zero when \(\lambda>b_0-a_0\). Differentiation under the integral is valid because its denominators are bounded away from zero on that fixed support.

Uniform integral cut comparison and row bounds

The preceding calculation has produced the finite-feature matrix \(\mathcal M\). We now collect its approximation error in the same block normalization as the original energy and record the row bounds needed when the endpoint types are sampled.

Finite Fourier inversion

Adding back the centering in Proposition 4 gives \[ \lim_{x\to\infty}\frac1x\sum_{1\le n<x} \overline{g_x(n)}f_x(n+1)=\overline{\mu_g}\mu. \tag{99}\] Here and throughout this subsection, \(x\) runs through positive integers. Indeed, the mean of \(\overline{g_x(n)}\) tends to \(\overline{\mu_g}\) by Lemma 3 and its initial-segment extension. For \(m\le x\), write \[\boldsymbol b_x(m) =\bigl(\Omega_{\mathcal B_{k,x}}(m)\bigr)_{1\le k<J}.\] Every coordinate lies in \(\{0,\ldots,J\}\): each counted prime factor exceeds \(x^{1/J}\). Thus reduction modulo \(J+1\) is injective on the set of count vectors. Put \(\zeta_0=\exp(2\pi i/(J+1))\). For \(\boldsymbol r\in\{0,\ldots,J\}^{J-1}\) and such a count vector \(\boldsymbol b\), the exact identity \[\mathbf 1_{\{\boldsymbol b=\boldsymbol r\}} =\frac1{(J+1)^{J-1}} \sum_{\boldsymbol h\in\{0,\ldots,J\}^{J-1}} \zeta_0^{\boldsymbol h\cdot(\boldsymbol b-\boldsymbol r)}\] expresses each bin event as a finite linear combination of characters. The two phase vectors in (99) are independent choices, and complex conjugation simply permutes the available root-of-unity phases. Applying this identity at \(n\) and \(n+1\) proves factorization of the limiting expectations of every pair of fixed functions of their count vectors. Both marginal limits exist by Lemma 3; the shift of the averaging range changes a bounded marginal average by \(O(1/x)\).

If \(c,d\in(0,1)\) are rational, choose \(J\) so that both are grid points. For \(m\le x\), the condition \(P^+(m)\le x^c\) is exactly the vanishing of all counts in bins with lower endpoint at least \(x^c\). Hence the joint smoothness event with thresholds \(x^c,x^d\) has a limiting density equal to the product of its two marginal limits. The next calculation identifies those limits.

The Dickman marginal

We recover the classical marginal law, originating with Dickman and developed by Ramaswami and de Bruijn (Dickman 1930; Ramaswami 1949; de Bruijn 1951), from the factorial moments already computed for the bin counts.

The marginal law also supplies the endpoint behavior of the limiting distribution. It gives \(0\le\rho(u)\le1\) for \(u>1\), and the same bounds hold at \(u=1\) by definition. The delay equation makes \(\rho\) nonincreasing on \([1,\infty)\). Its limit at infinity must be zero: if that limit were \(L>0\), integrating \(\rho'(u)=-\rho(u-1)/u\le-L/u\) would contradict nonnegativity. Consequently \[ D(t)= \begin{cases} 0,&t\le0,\\ \rho(1/t),&0<t<1,\\ 1,&t\ge1 \end{cases} \tag{103}\] is a continuous distribution function, including at zero and one, and its probability measure has no atoms.

Fixed and moving thresholds

Combining the Fourier inversion with Lemma 40 gives, for every pair of rational \(c,d\in(0,1)\), \[ \lim_{\substack{x\to\infty\\x\in\mathbb N}}\frac1x \#\{1\le n<x:P^+(n)\le x^c,\ P^+(n+1)\le x^d\} =\rho(1/c)\rho(1/d). \tag{104}\] We first extend this fixed-scale law to arbitrary real exponents and real counting endpoints. Fix \(c,d\in(0,1)\) and rational exponents \(c_-<c<c_+\) and \(d_-<d<d_+\) in \((0,1)\). Given real \(X\), put \(x=\lfloor X\rfloor+1\). Then \(x/X\to1\), the integer ranges \(n\le X\) and \(n<x\) agree, and for all sufficiently large \(X\), \[x^{c_-}\le X^c\le x^{c_+},\qquad x^{d_-}\le X^d\le x^{d_+}.\] Thus the normalized count over \(1\le n\le X\) with thresholds \(X^c,X^d\) lies between \(x/X\) times the two normalized counts in (104) at the lower and upper rational exponents. Taking lower and upper limits through real \(X\), then letting the rational exponents approach \(c,d\), proves \[ \begin{split} &\lim_{X\to\infty}\frac1X \#\{2\le n\le X:P^+(n)\le X^c,\ P^+(n+1)\le X^d\}\\ &\qquad=D(c)D(d)=\rho(1/c)\rho(1/d) \qquad(0<c,d<1). \end{split} \tag{105}\] The omission of \(n=1\) changes at most one summand. In (105), the exponents are fixed and the limit is through all real \(X\).

This also gives the fixed-scale upper-tail independence law discussed by Erdős and Pomerance (Erdős and Pomerance 1978, 311): \[ \begin{split} &\lim_{X\to\infty}\frac1X \#\{2\le n\le X:P^+(n)>X^c,\ P^+(n+1)>X^d\}\\ &\qquad=(1-D(c))(1-D(d)) \qquad(0\le c,d\le1). \end{split} \tag{106}\] For interior exponents this follows by inclusion–exclusion from (105) and the two marginal laws. The marginal for \(n+1\) is the one in Lemma 40, since shifting its counting range changes only \(O(1)\) terms. Replacing either strict comparison in (106) by a weak one has the same limit. Indeed, for fixed \(c>0\), equality \(P^+(m)=X^c\) is impossible unless \(X^c\) is a prime \(p\); in that case every such \(m\) is divisible by \(p\), giving only \(O(X^{1-c}+1)=o(X)\) possibilities for \(m\le X+1\). At exponent zero, the upper-tail condition holds for every \(m\ge2\), and equality can occur only at \(m=1\). At exponent one, an upper-tail condition on \(m\le X+1\) has only \(O(1)\) possible integers. These observations give the endpoint cases of (106) as well. Its right-hand side is continuous on \([0,1]^2\) by (103).

The ordering probability