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Source. Theorem (the paper's only theorem, unnumbered), printed pp. 1151--1152 (physical PDF pp. 1--2), read on the page images; restated as inequality (3) on p. 1152. The proof occupies sections 2--6 (pp. 1154--1169) and was not read here.

Statement

Let $q=1/p$, where $p\in\mathbb Z\setminus\{0,\pm1\}$, and let

$$\zeta_q(2)=\sum_{n=1}^{\infty}\frac{q^n}{(1-q^n)^2} =\sum_{n=1}^{\infty}\frac{p^n}{(p^n-1)^2}. \qquad (2)$$

Then $\zeta_q(2)$ is irrational, and only finitely many pairs of integers $a,b$ satisfy

$$\Big|\zeta_q(2)-\frac ab\Big|\le|b|^{-4.07869375}.$$

In terms of the irrationality exponent $\mu(\alpha)=\inf{c\in\mathbb R:\ |\alpha-a/b|\le |b|^{-c}$ has finitely many solutions $a,b\in\mathbb Z\}$, the paper restates this as (3), $\mu(\zeta_q(2))\le4.07869374\ldots$. The two printed constants differ in the last digit; both are recorded as printed.

Specialization. For $p=2$, by the paper's (1),

$$\zeta_{1/2}(2)=\sum_{n=1}^{\infty}\frac{2^n}{(2^n-1)^2} =\sum_{n=1}^{\infty}\frac{\sigma(n)}{2^n},$$

the number of Problem 250; so that number is irrational with irrationality measure at most $4.07869374\ldots$.

Proof pointer

A $q$-analog of the Rhin--Viola group-structure method: rational linear forms in $1$ and $\zeta_q(2)$ from a $q$-hypergeometric construction (section 2), their arithmetic through cyclotomic denominators (sections 1 and 3), a transformation group acting on the parameters (section 4), asymptotics (section 5) and the measure (section 6, pp. 1168--1169). Section 7 (pp. 1170--1171) gives a second route, a $q$-analog of Apéry's sequence, which also yields the irrationality. Nothing of this was checked here.

Coverage

Statement read on the page images; proof not read. Relied on as a refereed publication (Zbl 1044.11067). The paper itself records (p. 1151) that the irrationality was established by Duverney and the transcendence by Nesterenko before it.

Bears on. #250: at $p=2$ the theorem proves the problem's number irrational, with irrationality measure at most $4.07869374\ldots$; the paper (p. 1151) credits the irrationality to Duverney before it.