Zudilin 2002 irrationality measure q analogue zeta 2

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theorem: States that for q the reciprocal of an integer other than 0 and plus or minus 1 the number zeta_q(2) is irrational and its irrationality measure is at most 4.07869374...; at 1/q = 2 this is the series of problem 250.

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W. Zudilin, On the irrationality measure for a q-analogue of ζ(2), Mat. Sb. 193 (2002), no. 8, 49--70 (Russian); English translation Sb. Math. 193 (2002), no. 8, 1151--1172, DOI 10.1070/SM2002v193n08ABEH000674; Zbl 1044.11067 (reviewer T. Rivoal).

The copy read for this card is the English translation from mathnet.ru (record https://www.mathnet.ru/eng/sm674): 22 physical pages, printed pp. 1151--1172 (physical p. $n$ is printed p. $1150+n$), with a usable text layer; the theorem was also checked on the page images. Provenance: fetched from https://www.mathnet.ru/php/getFT.phtml?jrnid=sm&paperid=674&what=fullteng&option_lang=eng on 2026-09-17 (UTC), 274,626 bytes. The Russian original was not compared. That copy prints "Sbornik: Mathematics 193:8 1151–1172 ©2002 RAS(DoM) and LMS" in the header of its first page (checked on the page image; the text layer renders the copyright sign as "c⃝") and no license wording on its 22 pages, every other right reserved.

Contents

Definitions (1), p. 1151, for complex $|q|<1$: $\zeta_q(1)=\sum_{n\ge1}q^n/(1-q^n)=\sum_{n\ge1}\sigma_0(n)q^n$ and $\zeta_q(2)=\sum_{n\ge1}q^n/(1-q^n)^2=\sum_{n\ge1}\sigma_1(n)q^n$. The history on p. 1151 credits the irrationality of the $q$-harmonic series $\zeta_q(1)$ for $q=1/p$, $p\in\mathbb Z\setminus\{0,\pm1\}$, to Bézivin [1] and, independently, Borwein [2], and the irrationality of $\zeta_q(2)$ for the same $q$ to Duverney [3]; it then notes that Nesterenko's general theorem [4] on the arithmetic of values of modular functions already gives the transcendence of $\zeta_q(2)$ for every algebraic $q$ with $0<|q|<1$. Here [3] is Duverney, C. R. Acad. Sci. Paris Sér. I Math. 321 (1995), 1287--1289 (card) and [4] is Nesterenko, Mat. Sb. 187:9 (1996), 65--96 (card).

• Theorem (pp. 1151--1152; proof in sections 2--6, pp. 1154--1169): for $q=1/p$, $p\in\mathbb Z\setminus\{0,\pm1\}$, $\zeta_q(2)$ is irrational and $|\zeta_q(2)-a/b|\le|b|^{-4.07869375}$ has finitely many integer solutions; restated as (3) $\mu(\zeta_q(2))\le4.07869374\ldots$.
• p. 1152: "Nesterenko's Theorem 2 in [5]" gives $|\zeta_q(2)-a/b|>|b|^{-\gamma\ln^9\max\{2,\ln|b|\}}$ for all $a,b\in\mathbb Z$ and rational $q$, $0<|q|<1$, with $\gamma=\gamma(q)$; [5] is Nesterenko, Trudy Mat. Inst. Steklov. 218 (1997), 299--334 (Proc. Steklov Inst. Math. 218 (1997), 294--331), not the 1996 Mat. Sb. paper. The present theorem is called a qualitative improvement: a Liouville-type, finite measure.
• Method: a $q$-analog of the Rhin--Viola group-structure approach to $\mu(\zeta(2))$; section 1 $q$-arithmetic, section 2 the $q$-hypergeometric construction, section 3 arithmetic of the linear forms, section 4 the group structure, section 5 evaluation, section 6 (pp. 1168--1169) the measure. Section 7 (pp. 1170--1171) gives a $q$-analog of Apéry's sequence, which also proves $\zeta_q(2)$ irrational of Liouville type for $q^{-1}\in\mathbb Z\setminus\{0,\pm1\}$, answering Van Assche's question on a proof "in the spirit of Apéry" (p. 1152).

Compiled scope

The theorem's statement 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. It gives a proof of the irrationality of the problem's number later than Duverney 1995 and Nesterenko 1996, together with a Liouville-type irrationality measure, which the paper (p. 1152) sets against the weaker estimate it attributes to Nesterenko's Theorem 2 in [5].

Bears on. #250: at $p=2$ the theorem proves the problem's number $\sum_{n\ge1}\sigma(n)/2^n$ irrational, with irrationality measure at most $4.07869374\ldots$; the paper (p. 1151) credits the irrationality to Duverney [3] before it.

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