Postelmans 2007 irrationality zeta q 1 zeta q 2

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theorem_1_1: States that for q the reciprocal of an integer p at least 2 the explicit integers alpha_n, beta_n give nonzero forms beta_n zeta_q(1) - alpha_n whose n^2-th roots tend to at most p^(-3(pi^2-4)/pi^2), so zeta_q(1) is irrational.

theorem_1_2: States that for q the reciprocal of an integer p at least 2 the explicit integers a_n, b_n give nonzero forms b_n zeta_q(2) - a_n whose n^2-th roots tend to at most p^(-3(pi^2-8)/pi^2), so zeta_q(2) is irrational.

theorem_1_3: States that for q the reciprocal of an integer at least 2 the numbers 1, zeta_q(1) and zeta_q(2) are linearly independent over the rationals; at 1/q = 2 the last number is the divisor-sum series of problem 250.

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K. Postelmans and W. Van Assche, Irrationality of ζ_q(1) and ζ_q(2), J. Number Theory 126 (2007), no. 1, 119--154, DOI 10.1016/j.jnt.2006.11.011; Zbl 1138.11027 (reviewer W. Zudilin); arXiv:math/0604312.

The copy read for this card is arXiv:math/0604312v1 (stamped 13 April 2006; 34 pages; footer "Preprint submitted to J. Number Theory"), with a text layer; the theorems were also checked on the page images. Provenance: fetched from https://arxiv.org/pdf/math/0604312 on 2026-09-17 (UTC), 292,893 bytes. The journal version was not compared; result labels and pages below are those of the arXiv version. The arXiv record carries no license field, so arXiv's assumed license applies (arXiv:math/0604312), every other right reserved.

Contents

The $q$-zeta values are $\zeta_q(s)=\sum_{n\ge1}n^{s-1}q^n/(1-q^n)$ for $|q|<1$ and $s=1,2,\ldots$ (1.1), with $\lim_{q\to1}(1-q)^s\zeta_q(s)=(s-1)!\,\zeta(s)$ for $s\ge2$ (1.2). The introduction (p. 2) records: $\zeta_q(1)$ irrational for $q=1/p$, $p>1$ an integer; "Results of Nesterenko [12] show that $\zeta_q(2)$ is transcendental for every algebraic number $q$ with $0<|q|<1$. Zudilin gave an upper bound for the measure of irrationality of $\zeta_q(2)$ [22] with $1/q\in\{2,3,4,\dots\}$"; Krattenthaler, Rivoal and Zudilin [11] on $\zeta_q(2n)$ and $\zeta_q(2n+1)$. Standing convention from p. 2 on: "we only use values of $q$ for which $p=1/q\in\mathbb N\setminus\{0,1\}$".

• Theorem 1.1 (p. 2; proof pp. 15--17): for $q=1/p$, $p>1$ an integer, integers $\alpha_n,\beta_n$ given by (4.3)--(4.4) satisfy $\beta_n\zeta_q(1)-\alpha_n\ne0$ and $\lim|\beta_n\zeta_q(1)-\alpha_n|^{1/n^2}\le p^{-3(\pi^2-4)/\pi^2}<1$. (The printed statement has $\beta_n\zeta_q(2)-\alpha_n\ne0$, a misprint for $\zeta_q(1)$, as the displayed limit and section 4 show.)
• Theorem 1.2 (p. 2; proof pp. 23--28): integers $a_n,b_n$ given by (5.2)--(5.3) satisfy $b_n\zeta_q(2)-a_n\ne0$ and $\lim|b_n\zeta_q(2)-a_n|^{1/n^2}\le p^{-3(\pi^2-8)/\pi^2}<1$. With Lemma 1.1 (p. 3) this gives the irrationality of $\zeta_q(2)$, with the measure bound $3\pi^2/(\pi^2-8)\approx15.8369$ (p. 29; quoted on p. 3), weaker than Zudilin's $4.07869374\ldots$; for $\zeta_q(1)$ the analogous bound is $3\pi^2/(\pi^2-4)\approx5.04443$ (p. 19).
• Theorem 1.3 (p. 3; proof in section 6, pp. 29--33): the numbers $1$, $\zeta_q(1)$ and $\zeta_q(2)$ are linearly independent over $\mathbb Q$. The tool is Lemma 1.2 (p. 3), a linear independence criterion from simultaneous approximations with a common denominator.
• Sections 2--5: Hermite--Padé approximation to two Markov functions, multiple little $q$-Jacobi polynomials (Theorems 5.1 and 5.2), and the asymptotics of the approximants.

Compiled scope

Theorems 1.1--1.3 were read on the page images and are compiled as statements with proof pointers (Theorem 1.1, Theorem 1.2, Theorem 1.3), with the definitions of the approximants and the measure bounds; read status claims checked, the proofs not checked. Lemmas 1.1 and 1.2 are stated on the pages that use them. Relied on as a refereed publication; it is a later independent proof of the irrationality of the number of Problem 250.

Bears on. #250: Theorem 1.3 at $p=2$ gives the irrationality of $\zeta_{1/2}(2)=\sum_{n\ge1}\sigma(n)/2^n$ as part of a linear independence statement, and Theorem 1.2 with Lemma 1.1 gives it alone; section 5.4 derives from Theorem 1.2 an irrationality measure bound. #257: Theorem 1.1 with Lemma 1.1 at $p=2$ gives the irrationality of $\zeta_{1/2}(1)=\sum_{n\ge1}1/(2^n-1)$, the problem's sum for $A=\mathbb N$ only, a case Erdős settled in 1948.

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