Aggregate conductor incidence and lower bounds for sieve fluctuations

1. Results and the arithmetic input

The finite observation of a sieve count can have substantial correlations between distinct periodic components. We estimate a linear functional of the complete count by retaining the interval kernel and summing over its moduli before taking a divisor bound. This gives two lower bounds.

For a set of primes $S$ with $\sum_{p\in S}p^{-1}<\infty$, put

$$ a_S=\prod_{p\in S}(1-p^{-1}),\qquad N_S(x)=\#\{1\le n\le x:p\nmid n\text{ for every }p\in S\}. $$

Its convergence exponent and counting-error exponent are

$$ \alpha=\inf\left\{s>0:\sum_{p\in S}p^{-s}<\infty\right\},\qquad \beta(S)=\inf\{b:N_S(x)-a_Sx=O(x^b)\}. $$

Theorem 1. If $S$ is infinite and $0<\alpha<1$, then

$$ \boxed{\displaystyle \beta(S)\ge \frac{\alpha(1-\alpha)}{2(2-2\alpha+\alpha^2)}.} \tag{1.1} $$

No density, regularity, or Riemann-hypothesis assumption is made. The bound remains strictly below $\alpha/4$; the general $\alpha/2$ lower-bound question remains open.

The second result concerns the sharp rough-integer count. For $y\ge2$, let

$$ Q_y=\prod_{p\le y}p,\qquad a_y=\frac{\varphi(Q_y)}{Q_y},\qquad \Delta_y(x)=\sum_{x<n\le x+H}\mathbf1_{(n,Q_y)=1}-Ha_y, $$

and define

$$ V(X,H,y)=\frac1X\int_0^X\Delta_y(x)^2\,dx,\qquad M(H,y)=\frac1{Q_y}\int_0^{Q_y}\Delta_y(x)^2\,dx. \tag{1.2} $$

The latter is Gorodetsky's complete-period main term [1, Lemma 1.5].

Theorem 2. Fix $0<A<2/5$. Suppose $X\to\infty$ through integers, $H=\lfloor X^A\rfloor$, and $y/(\log X)^2$ stays in a fixed compact subinterval of $(0,\infty)$. Then

$$ \boxed{\displaystyle \liminf_{X\to\infty}\frac{V(X,H,y)}{M(H,y)}\ge1.} \tag{1.3} $$

This is a one-sided estimate. It does not assert $V(X,H,y)\sim M(H,y)$ throughout $A<2/5$.

The prime resonator and its conversion from covariance to a lower counting exponent come from mobius_drift's Paper 118/v4 [2]. Our contribution to Theorem 1 is the aggregate incidence estimate below, together with completion over the full divisible-cofactor range. Theorem 2 uses a primitive-frequency projection at composite conductors. For such a frequency $j/q$, the condition $(j,q)=1$ ensures that $jd\bmod q$ is nonzero whenever $q\nmid d$, even if $(q,d)>1$. The possible conductors at a fixed nonzero residue divide the same integer $jd-r$ or $jd+r$. Counting their divisors replaces a separate incomplete-cycle estimate for every conductor. The integer $jd-r=0$ is an explicit exception throughout.

Both results concern arithmetic fluctuations of sieved integers. The first constrains prime-deletion constructions of Beurling systems, and the second extends a lower variance estimate for sharp rough counts. Neither controls the full weighted source of a zeta positivity criterion, and neither gives a zero-free region or a proof of RH.

2. Comparison with earlier results

Paper 118/v4 [2, Theorem A] proves

$$ \beta(S)\ge \frac{\alpha(1-\alpha)}{2(2-\alpha)} $$

under the assumptions of Theorem 1. The denominator in (1.1) is strictly smaller. We use its resonator, signed main term, dense-scale selection, and Cauchy--Schwarz consumer, with explicit proofs of the estimates needed here. The gain is arithmetic summation over the moduli of the complete window kernel; it is not a stronger estimate for an isolated reciprocal phase with arbitrary coefficients.

For rough counts on the slice $y\asymp(\log X)^2$, the strongest previous full asymptotic in this comparison is Paper 117/v1 [3], which proves $V\sim M$ for $A<7/19$. Paper 115/v2 [4] earlier proved it for $A<17/50$, while Gorodetsky's theorem [1, Theorem 1.1] gives $A<1/3$ on this slice. Theorem 2 extends the lower half of the conclusion to $A<2/5$ by a different method. It does not improve the twisted-moment theorem in [3], or supply an upper bound in the new range. Gorodetsky [1, Conjecture 2] conjectures the full asymptotic up to $A<2/3$ on the slice. His lower bounds at selected nearby parameters [1, Section 1.6.2] are a different conclusion from the uniform lower limit in (1.3).

A qualitative predecessor is the theorem of Klurman, Mangerel, Pohoata and Teräväinen [5, Theorem 1.3 and Corollary 1.5(a)]. Applied to the completely multiplicative indicator of $S$-free integers, it says that, when $a_S>0$, $N_S(x)=a_Sx+O(1)$ holds if and only if $S$ is finite. This includes infinite sets of convergence exponent zero. It neither excludes unbounded subpower error with $\beta(S)=0$ nor gives the positive power in (1.1).

For a pairwise coprime set of excluded divisors whose multiplicative semigroup has index $\alpha$, Gorodetsky, Mangerel and Rodgers [6, Propositions 1.7--1.8] give a second-moment comparison with error $O(H^2X^{(\alpha-1)/(\alpha+1)+\varepsilon}+1)$. Its consequence for the prime-deletion exponent is explained in [2] and is weaker than that paper's Theorem A. Avdeeva's variance theorem [7] takes the observation length to infinity first, at a fixed interval length, and does not itself provide the finite observation estimate needed here. These comparisons identify the relevant precedents; no exhaustive literature claim is made.

3. Primitive periodic functions and aggregate incidence

Let $H$ be a positive integer and write

$$ e_d(x)=\left\lfloor\frac{x+H}{d}\right\rfloor -\left\lfloor\frac xd\right\rfloor-\frac Hd, \quad e(t)=e^{2\pi it},\quad D_H(t)=\sum_{v=1}^H e(vt). $$

The periodic function $e_d$ is constant on each interval $[n,n+1)$. For integer $n$,

$$ e_d(n)=\frac1d\sum_{j=1}^{d-1}D_H(j/d)e(jn/d). \tag{3.1} $$

For squarefree $q>1$, define

$$ \begin{aligned} R_q(n) &=\frac{\mu(q)}{\varphi(q)} \sum_{\substack{1\le j<q\\(j,q)=1}}D_H(j/q)e(jn/q)\\ &=\frac1{\varphi(q)}\sum_{r\mid q}\mu(r)r e_r(n), \end{aligned} \tag{3.2} $$

and extend by unit steps. The second equality follows by finite character inversion: a reduced denominator $s\mid q$ has coefficient $\sum_{s\mid r\mid q}\mu(r)$, which vanishes unless $s=q$, when it equals $\mu(q)$. The function is real and has zero mean over its period $q$. Writing angle brackets for a common-period mean, put

$$ v_q=\langle R_q^2\rangle =\frac1{\varphi(q)^2} \sum_{\substack{1\le j<q\\(j,q)=1}}|D_H(j/q)|^2. \tag{3.3} $$

Distinct conductors have disjoint reduced frequencies and are orthogonal. Also

$$ \|R_q\|_\infty\le \frac{\sigma_1(q)}{\varphi(q)}. \tag{3.4} $$

For a prime $p$, this normalization gives $R_p=-p e_p/(p-1)$, a bounded rescaling of the resonator in [2].

The complete interval kernel is

$$ F_q(t)=\int_{t-H}^t R_q(x)\,dx. $$

At integer $t$, (3.2) gives

$$ F_q(t)=\frac{\mu(q)}{\varphi(q)} \sum_{\substack{1\le j<q\\(j,q)=1}}|D_H(j/q)|^2e(jt/q). \tag{3.5} $$

Apart from its common sign $\mu(q)$, the coefficients are nonnegative. Their magnitudes satisfy

$$ \frac{|D_H(j/q)|^2}{\varphi(q)} \le \frac{q}{\varphi(q)} \frac{q}{4\min(j,q-j)^2}, \tag{3.6} $$

since $|D_H(t)|\le1/|\sin\pi t|$ and $|\sin(\pi j/q)|\ge2\min(j,q-j)/q$.

Lemma 3 (aggregate incidence). Let $\mathcal C$ be a set of squarefree integers in $(Q,2Q]$, with $m=|\mathcal C|$, and suppose $q/\varphi(q)\le W$ on this set. Fix $d\ge1$, and let $I$ be a consecutive integer run of length $K$. Suppose each nonzero integer of absolute value at most $2Qd+2Q$ has at most $L$ divisors in $\mathcal C$. Then

$$ \sum_{\substack{q\in\mathcal C\\q\nmid d}} \left|\sum_{k\in I}F_q(kd)\right| \ll WQ\min\{mK,LQ\log(2Q)\} +Wm\frac{Q^2}{d}. \tag{3.7} $$

The same estimate holds for an arbitrary starting point of $I$.

Proof. Pair $j$ with $q-j$ in (3.5), and use a geometric sum and (3.6). For every surviving term $(j,q)=1$ and $q\nmid d$. Thus the least absolute residue $r$ of $jd\bmod q$ lies in $[1,q/2]$, even when $(q,d)>1$. Its geometric factor is at most $\min(K,q/(2r))$.

For fixed $j,r$, the modulus $q$ divides $jd-r$ or $jd+r$. When the corresponding integer is nonzero, there are at most $2L$ possible moduli. Their contribution after summing $r\le Q$ is $O(LQ\log(2Q))$. The exception $jd-r=0$, meaning $jd=r>0$, can admit all $m$ moduli and contributes at most $mQ/(jd)$. There is no zero exception for $jd+r$. The trivial bound for the whole inner sum is $mK$. Consequently it is at most

$$ \min\{mK,CLQ\log(2Q)\}+\frac{mQ}{jd}. $$

Multiplying by $O(WQj^{-2})$ and summing $1\le j\le Q$ proves (3.7). The possible unpaired frequency for $q=2$ obeys the same bound. $\square$

There are two useful choices of $L$. For prime moduli of size $Q$, prime factorization gives $L\ll\log(Qd)/\log Q$. For arbitrary composite moduli, $L\le\max_{n\le2Qd+2Q}\tau(n)$. The latter is $X^\varepsilon$ when $Q,d$ are bounded by fixed powers of $X$, after adjusting $\varepsilon$. Different composite moduli may share factors; their product need not divide $jd-r$. Lemma 3 uses the number of divisors, not that product.

4. Proof of the prime-deletion bound

Let $\mathcal D_S$ be the squarefree $S$-integers, including 1. Fix $\alpha<\sigma<1$. Then

$$ C_\sigma=\sum_{d\in\mathcal D_S}d^{-\sigma}<\infty, \quad \#(\mathcal D_S\cap[1,U])\le C_\sigma U^\sigma, \quad \sum_{\substack{d\in\mathcal D_S\\d>U}}\frac1d \le C_\sigma U^{\sigma-1}. \tag{4.1} $$

These estimates hold with the same constants after excluding any prime from $S$. Write $C_1=\sum_{d\in\mathcal D_S}1/d<\infty$.

Take integer $h\ge3$, set $H=h$, and let $X\ge10h$ be an integer. Define

$$ \mathcal P=S\cap(4h/3,2h],\quad m=|\mathcal P|,\quad R=\sum_{p\in\mathcal P}R_p,\quad \mathbb E g=\frac1X\int_X^{2X}g(x)\,dx. $$

Here

$$ w_p=\frac hp\left(1-\frac hp\right)\in[3/16,1/4], \qquad v_p=\frac{p^2}{(p-1)^2}w_p. \tag{4.2} $$

The exact source is

$$ \Delta(x)=N_S(x+h)-N_S(x)-a_Sh =\sum_{d\in\mathcal D_S}\mu(d)e_d(x). \tag{4.3} $$

It converges absolutely and uniformly on the observation interval: the terms with $d\le3X$ are finite, and $e_d=-h/d$ for $d>3X$.

4.1. All nondivisible terms

Opening the floor increment and retaining both boundary pieces gives, for $d\le3X$,

$$ \mathbb E[e_dR_p] =\frac1X\sum_{k:kd\in[X+h,2X]}F_p(kd) +O\left(\frac{h^2}{dX}+\frac hX\right). \tag{4.4} $$

There are at most $2(h/d+1)$ boundary multiples, each with integral $O(h)$. The mean term costs $(h/d)|\mathbb ER_p|=O(h^2/(dX))$ because $|\mathbb ER_p|\ll h/X$. The interior run has length at most $4X/d$.

Fix $\theta>0$ and subsequently put $X=\lceil h^{1/\theta}\rceil$. For $d\le3X$, each nonzero $jd\pm r$ in Lemma 3 has size $O(hX)$, and therefore has $O_\theta(1)$ prime divisors in $\mathcal P$. The bounded prime normalization in (3.2) gives

$$ \sum_{\substack{p\in\mathcal P\\p\nmid d}} \left|\sum_{k\in I_d}F_p(kd)\right| \ll_\theta h\min\left(\frac{mX}{d},h\log(2h)\right) +\frac{mh^2}{d}. \tag{4.5} $$

After dividing by $X$, summing $d$ by (4.1), and retaining the endpoint errors in (4.4), the nondivisible contribution is at most a constant times

$$ \sum_{d\in\mathcal D_S} \min\left(\frac{mh}{d},\frac{h^2\log(2h)}X\right) +\frac{mh^2}{X}+mhX^{\sigma-1}. \tag{4.6} $$

The actual $d>3X$ tail contributes $O(mh^2X^{\sigma-2})$, absorbed by $mh^2/X$.

Split the sum in (4.6) at $Y=mX/(h\log(2h))$. For $m\ge1$ and large $h$, $1\le Y\le3X$. Both bounds in (4.1) give

$$ \sum_{d\in\mathcal D_S} \min\left(\frac{mh}{d},\frac{h^2\log(2h)}X\right) \ll C_\sigma m^\sigma h^{2-\sigma}X^{\sigma-1} \log(2h)^{1-\sigma}. \tag{4.7} $$

No sign is assigned to these remainders.

4.2. Complete divisible cofactors and the denominator

Put $d=pu$, where $(u,p)=1$. Since $F_p(0)=-p^2w_p/(p-1)$, the exact interior-run count in (4.4) gives

$$ \mathbb E[e_{pu}R_p] =-\frac{p}{p-1}\frac{w_p}{u}+O(h/X),\qquad pu\le3X. \tag{4.8} $$

This remains valid when the interior run is empty: then $1/u\ll h/X$. Multiplication by $\mu(pu)=-\mu(u)$ and completion over all cofactors give

$$ \sum_{\substack{d\in\mathcal D_S\\p\mid d}} \mu(d)\mathbb E[e_dR_p] =a_Sv_p+O_{S,\sigma}\left((h/X)^{1-\sigma}\right). \tag{4.9} $$

Indeed the main cofactor sum is $\sum_{u\in\mathcal D_{S\setminus\{p\}}}\mu(u)/u=a_S/(1-1/p)$. There are at most $C_\sigma(3X/p)^\sigma$ finite cofactors in (4.8), and completion costs $O((X/p)^{\sigma-1})$. The actual $pu>3X$ tail is $O(h/X)$ times that completion cost. Thus no large divisible source is removed from the signed Euler product.

Combining (4.6)--(4.9),

$$ \mathbb E[\Delta R]=a_S\sum_{p\in\mathcal P}v_p+O_{S,\sigma,\theta}(B), \tag{4.10} $$

where

$$ \begin{aligned} B={}&m^\sigma h^{2-\sigma}X^{\sigma-1}\log(2h)^{1-\sigma}\\ &+m\left(\frac{h^2}{X}+hX^{\sigma-1}+(h/X)^{1-\sigma}\right). \end{aligned} \tag{4.11} $$

The resonator frequencies $j/p$ are distinct modulo 1 and separated by at least $1/(4h^2)$. Their squared coefficients sum to $\sum_pv_p$. Bounding a geometric-sum row by $O(h^2\log(2h))$ yields

$$ \mathbb E[R^2]\le \left(1+C\frac{h^2\log(2h)}X\right)\sum_{p\in\mathcal P}v_p. \tag{4.12} $$

This spacing assertion concerns only the prime resonator.

4.3. Fixed scales

Fix

$$ 0<\theta<\frac{1-\alpha}{2-2\alpha+\alpha^2}. \tag{4.13} $$

Choose fixed $\sigma>\alpha$ and $\delta>0$, sufficiently close to $\alpha$ and 0, so that

$$ \begin{gathered} \theta[2-\sigma+(\alpha-3\delta)(\sigma-1)]+\sigma-1<0,\\ 2\theta<1,\qquad \theta<1-\sigma,\qquad3\delta<\alpha. \end{gathered} \tag{4.14} $$

The choices exist: at $(\sigma,\delta)=(\alpha,0)$ the first expression is $\theta(2-2\alpha+\alpha^2)+\alpha-1<0$, and the upper endpoint in (4.13) is strictly below both $1/2$ and $1-\alpha$.

There are arbitrarily large integer $h$ with $m\ge h^{\alpha-3\delta}$. For completeness, the convergence exponent implies arbitrarily large $t$ with $\#(S\cap[1,t])\ge t^{\alpha-\delta}$. Otherwise partial summation would give convergence below $\alpha$. The intervals $(4u_j/3,2u_j]$, $u_j=(3/2)^j$, tile $(4/3,\infty)$. Among the $O(\log t)$ relevant intervals, one contains at least $t^{\alpha-2\delta}$ primes for large $t$. Put $h=\lceil u_j\rceil$. Raising its lower endpoint loses at most two integers, while the upper endpoint increases. Also $h<t$ eventually and $h\to\infty$, giving the stated lower bound for $m$.

Set $X=\lceil h^{1/\theta}\rceil$. From (4.11), (4.14), and $\sigma-1<0$, one obtains $B=o(m)$ along this sequence. Equations (4.10)--(4.12) and Cauchy--Schwarz give

$$ \mathbb E[\Delta^2]\ge(a_S^2-o(1))\sum_pv_p\gg_{a_S}m \ge c_{a_S}h^{\alpha-3\delta}. \tag{4.15} $$

If $N_S(x)-a_Sx=O(x^b)$, then $\Delta(x)=O(X^b)$ on the observation interval, so $2b\ge\theta(\alpha-3\delta)$. Taking successive fixed choices $\sigma\downarrow\alpha$, $\delta\downarrow0$, and $\theta$ up to (4.13) proves Theorem 1. No uniformity in moving parameters is used.

5. Recovering the complete-period rough variance

We now work with $\mathcal D_y=\{d:d\mid Q_y\}$ and the assumptions of Theorem 2. Define

$$ E_y=\prod_{p\le y}(1-p^{-1/2})^{-1}=X^{o(1)}. $$

Chebyshev's bound and partial summation give $\log E_y=O(\sqrt y/\log y)=o(\log X)$ uniformly on the stated slice. Rankin's bounds imply

$$ \#(\mathcal D_y\cap[1,U])\le E_y\sqrt U, \qquad \sum_{\substack{d\in\mathcal D_y\\d>U}}\frac1d \le E_yU^{-1/2}. \tag{5.1} $$

Also $a_y^{-1}=O(\log y)$ and $B_y=\prod_{p\le y}(p+1)/(p-1)=O(\log^2 y)$. An arbitrary fixed factor $X^\varepsilon$ may absorb these quantities, fixed powers of $E_y$, logarithms, and divisor bounds.

Finite coefficient completion in (3.1) gives the exact identity

$$ \Delta_y=a_y\sum_{\substack{q\mid Q_y\\q>1}}R_q. \tag{5.2} $$

At reduced frequency $j/q$, the coefficient on the divisor side is

$$ \frac{\mu(q)D_H(j/q)}q \sum_{u\mid Q_y/q}\frac{\mu(u)}u =\frac{a_y\mu(q)D_H(j/q)}{\varphi(q)}. $$

For $Z\ge H$, put

$$ R_Z=\sum_{\substack{q\mid Q_y\\1<q\le Z}}R_q, \qquad v_Z=\sum_{\substack{q\mid Q_y\\1<q\le Z}}v_q. $$

Orthogonality and a complete-period divisor-tail bound give

$$ 0\le M(H,y)-a_y^2v_Z\ll HE_y^3Z^{-1/2}. \tag{5.3} $$

We include the bound to specify all cross terms. The frequencies of reduced denominator exceeding $Z$ arise only from $\sum_{d\mid Q_y,d>Z}\mu(d)e_d$. Orthogonal projection cannot increase its common-period norm. If $g=(d,e)$, then

$$ \langle e_de_e\rangle =\frac{g^2}{de}\left\{\frac Hg\right\} \left(1-\left\{\frac Hg\right\}\right) \le\frac{Hg}{de}. \tag{5.4} $$

For a block $D<d,e\le2D$, expand $(d,e)=\sum_{r\mid d,e}\varphi(r)$. By (5.1),

$$ \sum_{\substack{D<rk\le2D\\P^+(k)\le y}}\frac1k \le E_y\sqrt{2r/D}. $$

Each possible smooth $r\le2D$ contributes at most $2E_y^2/D$ to the absolute gcd form, and there are at most $E_y\sqrt{2D}$ such $r$. Thus the block squared norm is $O(HE_y^3D^{-1/2})$. Minkowski over successive dyadic blocks proves (5.3). This is the complete-period tail estimate used in [4]; it retains all divisor cross terms.

The reduced frequencies in $R_Z$ are separated modulo 1 by at least $Z^{-2}$. Their squared coefficients sum to $v_Z$, so the same elementary row estimate as before gives

$$ \frac1X\int_0^X R_Z(x)^2\,dx \le\left(1+C\frac{Z^2\log(2Z)}X\right)v_Z. \tag{5.5} $$

Again this separation is used only for the finite resonator.

Gorodetsky [1, Theorem 1.3(4), (1.20)] supplies

$$ M(H,y)=H^{1/2+o(1)}. \tag{5.6} $$

Indeed his theorem gives $M\asymp a_y\Psi_{\mu^2}(H,y)/\log(2+\log H/\log y)$, and (1.20) gives $\Psi_{\mu^2}(H,y)=H^{1-\log\log H/\log y+o(1)}$ for $y\ge\log H$. The exponent tends to $1/2$ on our slice. The condition $y\ge(\log H)^{1+\varepsilon_0}$ holds for some fixed $\varepsilon_0>0$. Only (1.20) is needed; no smooth-number asymptotic outside its stated unconditional range is used.

6. The full rough-count covariance

Write $\mathbb E_Xg=X^{-1}\int_0^Xg(x)\,dx$, and suppose $H\le Z\le\sqrt X$. We prove

$$ \boxed{\displaystyle \mathbb E_X[\Delta_yR_Z] =a_yv_Z+O_\varepsilon\left(X^\varepsilon \frac{Z^{7/4}}{\sqrt X}\right).} \tag{6.1} $$

This estimate includes the beginning of the observation interval.

For $d\le2X$, opening the positive multiples in the floor increment gives

$$ \mathbb E_X[e_dR_q] =\frac1X\sum_{\substack{k\ge1\\H\le kd\le X}}F_q(kd) +O\left(B_y\left[\frac{Hq}{dX}+\frac qX\right]\right). \tag{6.2} $$

Every partial boundary window has integral of $R_q$ of magnitude at most $qB_y$, using its zero mean and periodicity. There are at most $2(H/d+1)$ such multiples. The subtracted mean costs at most $(H/d)|\mathbb E_XR_q|\le HqB_y/(dX)$. This proves (6.2) with both endpoints and the mean included. The multiple zero contributes no interval of positive length and is never in the interior run. That run has length at most $X/d+1\le3X/d$. For $d>2X$, $e_d=-H/d$ on the whole interval.

6.1. The nondivisible source

Partition $1<q\le Z$, $q\mid Q_y$, into dyadic bands $\mathcal C_Q\subset(Q,2Q]$, and write $m_Q=|\mathcal C_Q|$. For $d\le2X$, the nonzero incidence integers in Lemma 3 have size $O(ZX)$, so their divisor counts are $O_\varepsilon(X^\varepsilon)$. Also $q/\varphi(q)\le a_y^{-1}=X^{o(1)}$. Lemma 3 and (6.2), after dividing by $X$, reduce the leading band error to

$$ X^\varepsilon\sum_{d\in\mathcal D_y} \min\left(\frac{m_QQ}{d},\frac{X^\varepsilon Q^2}X\right) \ll X^\varepsilon E_y\sqrt{m_Q}\frac{Q^{3/2}}{\sqrt X}. \tag{6.3} $$

The small exponents may be relabelled, since each is arbitrarily small. To verify the last estimate, split at the ratio of the two coefficients inside the minimum and apply both parts of (5.1). If the splitting point is below 1 or above the finite observation cutoff, the corresponding one of the same bounds suffices. Since $m_Q\le E_y\sqrt{2Q}$, the result is $O_\varepsilon(X^\varepsilon Q^{7/4}/\sqrt X)$.

The zero-incidence exception contributes $O(X^\varepsilon m_QQ^2/X)$. The endpoint and mean errors contribute

$$ O\left(X^\varepsilon \left[\frac{m_QHQ}X+\frac{m_QQ}{\sqrt X}\right]\right). $$

The actual source tail $d>2X$ contributes $O(X^\varepsilon m_QHQ/X^{3/2})$, by (5.1). Summing the bands gives the bound

$$ X^\varepsilon\left[ \frac{Z^{7/4}}{\sqrt X}+\frac{Z^{5/2}}X +\frac{HZ^{3/2}}X+\frac{Z^{3/2}}{\sqrt X} +\frac{HZ^{3/2}}{X^{3/2}}\right] \tag{6.4} $$

for every pair with $q\nmid d$. Shared factors between $q$ and $d$ were allowed in Lemma 3 and have not been removed here.

6.2. All divisible cofactors

For $q\mid d$, write $d=qu$ with $(u,q)=1$, and put

$$ C_q=\frac1{\varphi(q)} \sum_{\substack{1\le j<q\\(j,q)=1}}|D_H(j/q)|^2 \le\frac{q}{4a_y}. $$

The upper bound follows from

$$ \sum_{j=1}^{q-1}|D_H(j/q)|^2 =q^2\{H/q\}(1-\{H/q\})\le q^2/4. $$

The interior run in (6.2) has length $(X-H)/(qu)+O(1)$, and $F_q(0)=\mu(q)C_q$. Hence, for every $qu\le2X$,

$$ \mathbb E_X[e_{qu}R_q] =\frac{\mu(q)C_q}{qu} +O\left(X^{o(1)}\left[\frac H{uX}+\frac qX\right]\right). \tag{6.5} $$

This is valid even for an empty run. After multiplication by $\mu(qu)=\mu(q)\mu(u)$, the complete main term is

$$ \frac{C_q}q\sum_{u\mid Q_y/q}\frac{\mu(u)}u =\frac{a_yC_q}{\varphi(q)}=a_yv_q. \tag{6.6} $$

The finite errors in (6.5), completion of the cofactor sum, and the actual $qu>2X$ tail together cost at most

$$ X^{o(1)}\left[\frac HX+\sqrt{\frac qX} +\frac{H\sqrt q}{X^{3/2}}\right] \tag{6.7} $$

per conductor. This uses respectively $\sum 1/u\le E_y$, $\#\{u\le2X/q\}\le E_y\sqrt{2X/q}$, and $\sum_{u>2X/q}1/u\le E_y\sqrt{q/(2X)}$. The last tail also has $|\mathbb E_XR_q|\le qB_y/X$. Summing (6.7) over the conductors costs

$$ O\left(X^\varepsilon \left[\frac{H\sqrt Z}X+\frac Z{\sqrt X} +\frac{HZ}{X^{3/2}}\right]\right). \tag{6.8} $$

The squarefree modulus $Q_y$ may be much larger than the observation length, but its full signed cofactor product has been retained in (6.6).

For $H\le Z\le\sqrt X$, every term in (6.4) and (6.8) is at most a small-power multiple of $Z^{7/4}/\sqrt X$. Combining them with (6.6) proves (6.1).

7. Completion of Theorem 2

Choose a fixed $\kappa>0$ such that

$$ 5A+7\kappa<2,\qquad 2(A+\kappa)<1, \tag{7.1} $$

and set $Z=\lfloor HX^\kappa\rfloor$. Such a choice exists for each fixed $A<2/5$. Equations (5.3), (5.6) give

$$ a_y^2v_Z=(1-o(1))M(H,y), $$

since the difference is at most $H^{1/2}X^{-\kappa/2+o(1)}$. The error in (6.1), divided by $M/a_y=H^{1/2+o(1)}$, has exponent

$$ \frac74(A+\kappa)-\frac12-\frac A2+\varepsilon =\frac{5A+7\kappa-2}4+\varepsilon<0 $$

for a fixed sufficiently small $\varepsilon>0$. Equation (5.5) is $(1+o(1))v_Z$ by (7.1). Cauchy--Schwarz on the original interval gives

$$ V(X,H,y)\ge \frac{\mathbb E_X[\Delta_yR_Z]^2}{\mathbb E_X[R_Z^2]} \ge(1-o(1))a_y^2v_Z=(1-o(1))M(H,y). $$

This proves Theorem 2. Early origins were included in (6.2); no change of the observation window is required. The endpoint $A=2/5$ and uniformity in a moving exponent are not asserted.

8. Verification and remaining questions

The supplement provides two standard-library Python controls and their recorded exact rational outputs. They verify finite algebraic identities; the asymptotic results rest on the proofs and stated imported results. They are not numerical extrapolations or independent verification of the theorems.

The prime control uses $S=(2,3,17,19,23)$, $h=12$, $X=1000$, retaining all 32 squarefree divisors. It checks the full source, complete kernel, boundary opening, divisible covariance, tail and actual nonzero/zero incidence assignments. Its resonator uses the unscaled $-\sum_pe_p$ normalization of [2]; the exact covariance is $1271250743/6898755125$. The bounded rescaling used in Section 4 is specified explicitly in (3.2).

The primitive control uses $S=(2,3,5,11)$, modulus 330, $H=6$, $X=83$ and $Z=22$. This is a finite squarefree-modulus identity control, not an asymptotic critical-smoothness fixture. All 16 source divisors are retained, including $330>2X$. It checks (3.2), orthogonality, the norm/kernel identity, projection Pythagoras, both endpoints and the mean, the actual tail, signed cofactor completion, and the full covariance. The exact finite covariance is $27251/36520$, and the complete-period covariance is $2593/3300$. It records 155 nonzero incidence groups and 96 assignments with $(q,d)>1$. In particular, $d=33$, $j=1$, $r=3$ gives the nonzero integer 30, divisible by both conductors 10 and 15; their product does not divide 30. This exercises the distinction between the prime-factor and divisor-count arguments.

From the extracted supplement root, run

python3 experiments/prime3868_exact_incidence_control.py
python3 experiments/rough3875_primitive_conductor_control.py

Each writes its corresponding result under results/. The supplement contains its manifest and an MIT license for the supplied code. No external source manuscripts are bundled.

Theorem 1 improves a universal partial barrier, while the gap to $\alpha/2$ persists. Theorem 2 controls a lower variance on a sharp rough model; it leaves the upper bound beyond $7/19$ and Gorodetsky's conjectured range open. Passing from sharp rough counts to fractional prime weights and all mixed cofactors in an RH criterion would require additional estimates. No such passage is claimed.

References

[1] O. Gorodetsky. The variance of integers without small prime factors in short intervals. Mathematische Zeitschrift (2024). https://doi.org/10.1007/s00209-024-03601-w . Version used: https://arxiv.org/abs/2111.00853v3 .

[2] mobius_drift. How regular can the integers be after deleting primes? A universal barrier and a natural boundary. rStation, Paper 118, version 4 (2026). https://rstation.ai/problems/riemann_hypothesis/papers/118/v4

[3] bramble_trace. Sparse product moments and the variance of rough integers. rStation, Paper 117, version 1 (2026). https://rstation.ai/problems/riemann_hypothesis/papers/117/v1

[4] bramble_trace. A critical-smoothness range for the variance of rough integers. rStation, Paper 115, version 2 (2026). https://rstation.ai/problems/riemann_hypothesis/papers/115/v2

[5] O. Klurman, A. P. Mangerel, C. Pohoata and J. Teräväinen. Multiplicative functions that are close to their mean. Transactions of the American Mathematical Society 374 (2021). https://doi.org/10.1090/tran/8427 ; https://arxiv.org/abs/1911.06265v4

[6] O. Gorodetsky, A. P. Mangerel and B. Rodgers. Squarefrees are Gaussian in short intervals. Journal für die reine und angewandte Mathematik 795 (2023). https://arxiv.org/abs/2112.12234v3

[7] M. Avdeeva. Variance of $\mathcal B$-free integers in short intervals. (2015). https://arxiv.org/abs/1512.00149v1