Smet 2009 irrationality proof q extension zeta 2

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theorem_1_1: States that for q the reciprocal of an integer at least 2 the number zeta_q(2) is irrational and its irrationality measure is at most 10 pi^2/(5 pi^2 - 24), about 3.8936; at 1/q = 2 this is the series of problem 250.

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C. Smet and W. Van Assche, Irrationality proof of a q-extension of ζ(2) using little q-Jacobi polynomials, Acta Arith. 138 (2009), no. 2, 165--178, DOI 10.4064/aa138-2-5; Zbl 1226.11074 (reviewer P. Bundschuh); arXiv:0809.2501.

The copy read for this card is the arXiv version stamped "arXiv:0809.2501v3 [math.CA] 18 Sep 2008" (13 pages; the date line "November 15, 2018" under the authors is an artifact of the arXiv rendering), with a text layer; Theorem 1.1 was also checked on the page image. Provenance: fetched from https://arxiv.org/pdf/0809.2501 on 2026-09-17 (UTC), 174,380 bytes. The journal version was not compared; labels and pages are those of the arXiv version. The arXiv record names arXiv's non-exclusive distribution license (arXiv:0809.2501), every other right reserved.

Contents

The introduction (p. 1) defines $\zeta_q(s)=\sum_{k\ge1}k^{s-1}q^k/(1-q^k)$ and records the history for $q=1/p$, $p\in\mathbb N\setminus\{0,1\}$: the paper (arXiv v3, 2008) reports that only $\zeta_q(1)$ and $\zeta_q(2)$ had then been shown irrational for these $q$. For $\zeta_q(1)$ the paper credits Borwein [5],[6] in 1991 and, by a different method, Bundschuh and Väänänen [7] in 1994, and notes that a 1988 result of Bézivin [3] also yields this irrationality. For $\zeta_q(2)$ it credits the irrationality to Duverney [8] in 1995 and the transcendence, indeed that of every $\zeta_q(2s)$ with $s\in\mathbb N$, to a general result of Nesterenko [11]; it adds that Postelmans and Van Assche [12] proved $1,\zeta_q(1),\zeta_q(2)$ linearly independent over $\mathbb Q$. Here [8] is Duverney, C. R. 321 (1995), 1287--1289 (card), [11] Nesterenko, Mat. Sb. 187:9 (1996), 65--96 (card), [12] Postelmans and Van Assche, J. Number Theory 126 (2007), 119--154 (card) and [15] Zudilin, Mat. Sb. 193 (2002), no. 8, 49--70 (card).

• Theorem 1.1 (p. 2; proof concluded on p. 12): for $q=1/p$ with $p\in\mathbb N\setminus\{0,1\}$, $\zeta_q(2)$ is irrational and $|\zeta_q(2)-a/b|\le|b|^{-\rho}$ with $\rho=10\pi^2/(5\pi^2-24)$ has at most finitely many integer solutions; hence $2\le\mu(\zeta_q(2))\le10\pi^2/(5\pi^2-24)\approx3.8936$, "sharper than the upper bound 4.07869374 given by Zudilin [15]".
• Lemmas 1.2 and 1.3 (p. 2): the irrationality criterion from nonzero integer linear forms tending to zero, and, under the same hypotheses, the measure bound $\mu(x)\le1+1/s$ from $|b_nx-a_n|=O(b_n^{-s})$ with $b_n<b_{n+1}<b_n^{1+o(1)}$.
• Remark 1.4 (p. 2): for natural $r$ the difference $\sum_k kq^k/(1-q^k)-\sum_k kq^{rk}/(1-q^k)$ is rational, so $\sum_k kq^{rk}/(1-q^k)$ is irrational with the same measure.
• Method: type I Hermite--Padé approximation to two functions $f_1,f_2$ with $f_1(1)=\zeta_q(1)$, $f_2(1)=\zeta_q(2)$, the polynomials being little $q$-Jacobi polynomials (sections 2--3), a $q$-adaptation of the Apéry-type proofs for $\zeta(2)$.

Compiled scope

Theorem 1.1 was read on the page image and is recorded with its specialization to $p=2$; the proof was not read. Relied on as a refereed publication. At $p=2$ it proves again the irrationality of the number of Problem 250, which the paper credits to Duverney in 1995 (p. 1), and adds the measure bound.

Bears on. #250: Theorem 1.1 at $p=2$ proves the problem's number $\sum_{n\ge1}\sigma(n)/2^n=\zeta_{1/2}(2)$ irrational, with irrationality measure at most $10\pi^2/(5\pi^2-24)\approx3.8936$.

No file of this source is held: no license on record permits its redistribution, and the card cites the edition it names above.