Duverney 1993 proprietes arithmetiques serie fonctions theta

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theoreme_2: States that a q-adic series with integer coefficients bounded by r(n), where limsup r(n+1)/r(n) < |q|, and vanishing on a run of k places after n_k, with r(n_k+k+1)/|q|^k -> 0, has, if it is rational, its partial sum up to n_k exactly equal to the value for all large k.

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D. Duverney, Propriétés arithmétiques d'une série liée aux fonctions thêta, Acta Arith. 64 (1993), no. 2, 175--188; Zbl 0779.11028 (reviewer P. Bundschuh).

The retained folder-name PDF is the author's scan of the fourteen printed pages (head "ACTA ARITHMETICA LXIV.2 (1993)"; physical PDF p. $n$ is printed p. $174+n$). It has no text layer; the statements below were read on the page images. Provenance: fetched from https://danielduverney.fr/documents/theorie-des-nombres/acta1.pdf on 2026-09-17 (UTC), 407,533 bytes. The journal version was not compared. The scan is image-only and its rendered first and last pages show no copyright or license line; the journal's record offers the PDF under the download link "Pobierz zgodnie z CC-BY", rendered "Free download under CC-BY license" on the English site, and names no version or URL for it (https://www.impan.pl/get/doi/10.4064/aa-64-2-175-188, read 2026-10-02): the Creative Commons Attribution license, with no version stated.

Contents

• Théorème 1 (p. 176; proof in section 5): for $q\in\mathbb Z$, $|q|\ge2$, $x_q=\sum_{n\ge0}q^{-n^2}$ is not quadratic. Section 1 recalls that $x_q$ is irrational (its $q$-adic expansion is a nonperiodic sequence of $0$s and $1$s), Liouville's proof by partial sums, Bundschuh's result that $x_q$ is not a Liouville number, Borwein's theorem that $\sum_{n\ge1}(-1)^n/(q^n+r)\notin\mathbb Q$ for rational $r\ne-q^n$, and the relation $2x_q=1+\theta_3(0,\log q/(i\pi))$ (3). Page 176 also records Erdős's conjecture (the paper's [7]) that $\sum_k q^{-n_k}$ is not quadratic whenever $n_k>ck^2$. No catalog page is identified for $x_q$ here; the transcendence of $\theta_3(q)$ for algebraic $q$ later followed from Nesterenko's 1996 theorem (Corollaire 4 of Waldschmidt's exposé).
• Théorème 2 (p. 176; proof in section 2, p. 178): the irrationality criterion by partial sums for series $\sum_{n\ge0}a(n)q^{-n}$ with integer coefficients that vanish on runs of length $k$ after $n_k$. This is the tool of the Lemme of Duverney 1995.
• Théorèmes 3 and 4 (p. 176) are quoted from the literature (the paper's [10] and [6]): for $a(n)\in\mathbb N$ nonzero infinitely often with $\sum_{k\le n}a(k)=o(n)$, $\sum_{n\ge1}a(n)q^{-n}$ is irrational for every integer $q\ge2$; and for integers $a(1)<a(2)<\cdots$ with $a(n+1)-a(n)\to\infty$, $\sum_{n\ge1}a(n)q^{-a(n)}$ is irrational for every integer $q\ge2$.
• Plan (p. 178): section 2 proves Théorème 2; section 3 gives a pedagogical application; section 4 states and proves the technical lemma on the zeros of $\varrho(n)$; section 5 proves Théorème 1.

Compiled scope

Only Théorème 2 is extracted, with its claims checked on the page image and its half-page proof read and summarized. Théorème 1 and sections 3--5 were not read beyond the statements above. Nothing here is independently reviewed.

Bears on. #250, as the criterion consumed by the Lemme of Duverney 1995, whose Théorème settles the problem; the paper itself proves nothing about the problem's series.