Weighted divisibility moments

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These are proved estimates, not a resolution of Problem 25. They remove the overlap hypothesis from the parent-refinement obstruction on a substantial, but bounded, range of sampling scales.

A weighted version of harmonic domination

Let $W:\mathbb N\to[0,\infty)$ depend only on divisibility by a fixed finite set of primes, and suppose that $W$ is nondecreasing under divisibility. Regard the same function as a function on $\widehat{\mathbb Z}$, with Haar measure $\mu$. Then

$$\sum_{n\le X}\frac{W(n)}n \le H_{\lfloor X\rfloor}\int W\,d\mu. \tag{1}$$

Here the stated prime-divisibility dependence means that only the indicators $\mathbf1_{p\mid n}$, not unbounded valuations, are used. Thus $W$ is bounded. Each superlevel set $\{W>t\}$ is a multiples set generated by finitely many squarefree divisors of the product of the chosen primes. Apply the exact multiples harmonic inequality from basic reductions to every superlevel, and integrate in $t$. This proves (1).

The parent weights

Use the notation of the parent-refinement note:

$$\mathcal P=\{p:3\le p\le z\},\quad \lambda=\sum_{p\in\mathcal P}\frac1p,\quad r=\lfloor2\lambda-6\sqrt\lambda\rfloor,\quad b=\lceil2\lambda+2\sqrt\lambda\rceil,\quad s=b-r.$$

All assertions below concern sufficiently large $\lambda$. Put

$$P_n=\prod_{\substack{p\in\mathcal P\\p\mid n}}(1-2/p)^{-1}, \qquad c_0=2\log2-1, \qquad \kappa=2+c_0=1+2\log2.$$

For either $a=1/2$ or $a=1$, (1) and a weighted Chernoff bound give

$$\frac1{H_{\lfloor X\rfloor}} \sum_{\substack{n\le X\\\omega_{\mathcal P}(n)\ge r}} \frac{P_n^a}{n} \le \exp\{-c_0\lambda+O(\sqrt\lambda)\}. \tag{2}$$

Indeed $W(n)=P_n^a\mathbf1_{\omega_{\mathcal P}(n)\ge r}$ is divisibility-monotone. With independent Bernoulli variables $I_p$ of parameters $1/p$, its Haar expectation is at most

$$2^{-r}\prod_{p\in\mathcal P} \left(1-\frac1p+\frac2p(1-2/p)^{-a}\right).$$

For each of the two fixed values of $a$, the logarithm of the local factor is $1/p+O(p^{-2})$, with an absolute summable error, including the prime 3. The product is consequently $\exp\{\lambda+O(1)\}$. Substituting the definition of $r$ proves (2), uniformly in $X$.

An overlap-free protected-trace bound

Consider arbitrary distinct singleton moduli $q$, each prescribing 0 or 1 at at least $r$ primes of $\mathcal P$. Write

$$T=\{n:\#\{p\in\mathcal P:n\bmod p\in\{0,1\}\}\le b\}, \qquad B=\bigcup_q\{a_q+tq:t\ge1\},\quad 0\le a_q<q.$$

No preparatory sieve, incremental-footprint hypothesis, or overlap restriction is imposed. Uniformly in this family and $X\ge2$,

$$\frac{H_{T\cap B}(X)}{H_{\lfloor X\rfloor}} \le H_{\lfloor X\rfloor} \exp\{-\kappa\lambda+O(\sqrt\lambda\log(2\lambda))\} +\exp\{-c_0\lambda+O(\sqrt\lambda\log(2\lambda))\}. \tag{3}$$

To prove this, discard the classes prescribing successes at more than $b$ primes, since they miss $T$. Let $\theta_q=\mu(T\mid[a_q]_q)$, $\sigma_X=\sum_{q\le X}\theta_q/q$, and $H=H_{\lfloor X\rfloor}$. The already-proved dimension-two progression estimate, namely (9) of the parent-refinement note, states

$$\frac{H_{T\cap B}(X)}H \le C\sigma_X+ \frac{\exp\{O(\sqrt\lambda\log(2\lambda))\}}H \sum_{q\le X}\frac{\sqrt{P_q}}q. \tag{4}$$

Its hypotheses concern only these classes and $T$, and do not include a multiplicity bound. Formula (2) with $a=1/2$ bounds the second term by the second term in (3).

The same note gives

$$\theta_q\le V_zF_sP_q,\qquad V_z=\exp\{-2\lambda+O(1)\},\qquad F_s=\exp\{O(\sqrt\lambda\log(2\lambda))\}.$$

Every retained modulus has at least $r$ prime factors from $\mathcal P$. Distinctness permits its reciprocal sum to be bounded by the sum over all integers in (2). Using $a=1$,

$$\sigma_X\le V_zF_s H \exp\{-c_0\lambda+O(\sqrt\lambda)\}.$$

Substitution in (4) proves (3). In an infinite family only finitely many moduli $q\le X$ contribute. The representation $t\ge1$ is exactly the original activation $n\ge q$; no inactive root has been counted.

For each fixed $\varepsilon>0$, (3) tends to zero uniformly when

$$H_{\lfloor X\rfloor}\le \exp\{(\kappa-\varepsilon)\lambda\}. \tag{5}$$

Since $\lambda=\log\log z+O(1)$, this includes $X\le\exp((\log z)^{\kappa-\varepsilon})$, after reducing the fixed margin if necessary, and in particular $X\le z^A$ for every fixed $A$. The elementary two-sided prime reciprocal estimate used here is already established in the earlier notes; the formulation (5) needs no translation to $z$.

Thus the preparations with moduli at most $z$ cannot produce a fixed positive protected harmonic deletion in any fixed polynomial-in-$z$ epoch, even with unrestricted overlap and without assuming a small incremental footprint. Epochs starting at much larger scales are not excluded.

Improvement of the all-scale bounded-overlap error

The $a=1/2$ case also improves (11) of the parent-refinement note to

$$\frac{H_{T\cap B}(X)}H \le C\sigma_X+ \exp\{-c_0\lambda+O(\sqrt\lambda\log(2\lambda))\} \qquad(X\ge2). \tag{6}$$

This follows directly from (4); its old cutoff split and Cauchy--Schwarz loss are unnecessary. Combining (6) with that note's unchanged survivor comparison and in-parent multiplicity bound gives

$$\frac{H_{S\cap T\cap B}(X)}H \le L\exp\{O(\sqrt\lambda\log(2\lambda))\}\tau +\exp\{-c_0\lambda+O(\sqrt\lambda\log(2\lambda))\}. \tag{7}$$

The stronger error does not remove the multiplicity hypothesis from this all-scale, footprint-sensitive conclusion. Formula (3) does remove it, but only yields a vanishing bound in the range (5).