---

Statement

Setting (p. 2). Let $\theta:\mathbb N\to\mathbb R\cup\{\infty\}$ be an arithmetic function. $\mathcal B$ is the set of positive integers consisting of $n=1$ and every $n\ge2$ with prime factorization $n=p_1^{\alpha_1}\cdots p_k^{\alpha_k}$, $p_1<\cdots<p_k$, such that $p_{j+1}\le\theta(p_1^{\alpha_1}\cdots p_j^{\alpha_j})$ for $0\le j<k$, the empty product being $1$. The standing hypothesis is

$$\theta:\mathbb N\to\mathbb R\cup\{\infty\},\qquad \theta(1)\ge2,\qquad \theta(n)\ge P^+(n)\quad(n\ge2), \tag{3}$$

with $P^+(n)$ the largest prime factor of $n$. $B(x)$ counts the $n\le x$ in $\mathcal B$ and $\chi$ is the indicator function of $\mathcal B$. With $\theta(n)=\sigma(n)+1$, $\mathcal B$ is the set of practical numbers, by Sierpinski and Stewart (p. 2).

Lemma 1 (p. 2, recalled from the author's earlier papers). Under (3), for $\operatorname{Re}(s)>1$, $\sum_{n\ge1}\chi(n)n^{-s}\prod_{p\le\theta(n)}(1-p^{-s})=1$, and the equation also holds at $s=1$ if $B(x)=o(x)$.

Lemma 2 (p. 2). Let $\theta$ satisfy (3), let $q$ be prime and $h\in\mathbb N$. For $\operatorname{Re}(s)>1$,

$$\sum_{\substack{n\ge1\\ q^h\parallel n}}\frac{\chi(n)}{n^s} \prod_{p\le\theta(n)}\left(1-\frac1{p^s}\right) =\frac{1-1/q^s}{q^{sh}} \sum_{\substack{n\ge1\\ \theta(n)\ge q}}\frac{\chi(n)}{n^s} \prod_{p\le\theta(n)}\left(1-\frac1{p^s}\right),$$

and if $B(x)=o(x)$ the equation also holds at $s=1$. Here $p$ runs over primes and $q^h\parallel n$ means that $q^h$ divides $n$ and $q^{h+1}$ does not.

The paper calls this a new identity (p. 1) and states it, with Lemma 1, in the general setting because it applies to other sets than the practical numbers (p. 2).

Proof pointer

Pp. 2--3. Every integer $m$ splits uniquely as $m=nr$ with $n\in\mathcal B$ and every prime factor of $r$ above $\theta(n)$; summing $m^{-s}$ over the $m$ with $q^h\parallel m$ in two ways and dividing by $\zeta(s)$ gives the identity for $\operatorname{Re}(s)>1$ after Lemma 1. At $s=1$ the two sums are shown right-continuous, using the estimate $\prod_{p\le\theta(n)}(1-p^{-s})\asymp s-1+1/\log\theta(n)$ uniformly for $1\le s\le2$ from the author's Math. Comp. paper.

Read depth

Claims checked: the setting, (3), Lemma 1 and Lemma 2 were read on the page images of pp. 2--3 of arXiv version 3, and the proof was followed at the level of the sketch above. Nothing here is independently reviewed.

Dependencies

Lemma 1 (p. 2), taken from A. Weingartner, On the constant factor in several related asymptotic estimates, Math. Comp. 88 (2019), 1883--1902, Lemma 1, and A. Weingartner, A sieve problem and its application, Mathematika 63 (2017), 213--229, Theorem 1. Lemma 2 feeds Lemma 3 (p. 3) and through it Theorem 1.

Source. Andreas Weingartner, The constant factor in the asymptotic for practical numbers, arXiv:1906.07819; the edition read is named on the source card.

Bears on

No Erdős problem directly; the lemma is a tool for Theorem 1.