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Claim. Corollary 1.2 of Daniel Duverney and Yohei Tachiya, Refinement of the Chowla–Erdős method and linear independence of certain Lambert series, Forum Math. 31 (2019), no. 6, 1557--1566, concerns the sets $F_s(E)$: for a sequence $E=\{e_n\}$ of pairwise coprime integers $e_n>1$ with $e_n\le n^\mu$ for all large $n$ and some $\mu>1$, and for $2\le s\le\infty$, $F_s(E)$ is the set of finite products $\prod e_i^{m_i}$ with $0\le m_i<s$, with no bound on the exponents when $s=\infty$. The corollary states that for an integer $q$ with $|q|>1$, positive integers $h$ and $\ell$, and $L=\operatorname{lcm}(1,\ldots,\ell)$ with $|q|L\le s$ (no condition when $s=\infty$), the numbers

$$1,\qquad \sum_{n\in F_s(E)}\frac{1}{q^{jn^i}-1} \quad(1\le i\le\ell,\ 1\le j\le h)$$

are linearly independent over $\mathbb Q$. With $q=2$ and $h=\ell=1$ this says that $\sum_{n\in A}1/(2^n-1)$ is irrational for $A=F_s(E)$, an instance of Problem 257 answered yes; the paper's Example 1.1 is the squarefree integers, $F_2$ of the primes, and its Example 1.3 the integers coprime to a fixed modulus, $F_\infty$ of the other primes. The paper presents these as classes supporting the conjecture of Erdős and Graham for arbitrary increasing exponent sequences, not as its proof. The method is Theorem 1.1, a refinement of the Chowla–Erdős congruence construction: if $f(q)=\sum_n\theta(n)/q^n$ is rational and the integer coefficients $\theta(n)$ are divisible by $q^m$ along products of $m$ large generators of $E$ (hypothesis $(H_1)$) and have at most $n(2+\log n)^\nu$ absolute mass on every progression (hypothesis $(H_2)$), then $\theta$ vanishes infinitely often in every residue class. For $A=F_s(E)$ the coefficient $\theta(n)=\#\{a\in A:a\mid n\}$ has the divisibility $(H_1)$ by its product formulas (4.3)-(4.4) and the growth $(H_2)$ from $\theta\le d(n)$, while it is positive on the multiples of any element of $A$. The source card duverney_tachiya_2019_refinement_chowla_erdos_method_linear_independence_certain_lambert_series digests the authors' preprint and works out this specialization.

Covers. Every support $A=F_s(E)$ with $E$ as above and $2\le s\le\infty$, the squarefree integers and the integers coprime to a fixed modulus among them, at the base $2$ of the question and at every integer base $q$ with $1<|q|\le s$. Not covered: supports without this multiplicative structure, for which the divisibility hypothesis $(H_1)$ is not available.

Acceptance. Refereed: Forum Mathematicum, volume 31, issue 6 (2019), pp. 1557--1566, published online 14 August 2019 by the record of its DOI. The site does not cite the paper, so no reviewed evidence is listed. The proof is not checked here.

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