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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Diophantine properties for $ q$-analogues of Dirichlet's beta function at positive integers


Authors: Frédéric Jouhet and Elie Mosaki
Journal: Trans. Amer. Math. Soc. 363 (2011), 1533-1554
MSC (2010): Primary 11J72; Secondary 11M41, 33D15
Published electronically: October 15, 2010
MathSciNet review: 2737276
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Abstract: In this paper, we define $ q$-analogues of Dirichlet's beta function at positive integers, which can be written as $ \beta_q(s)=\sum_{k\geq1}\sum_{d\vert k}\chi(k/d)d^{s-1}q^k$ for $ s\in\mathbb{N}^*$, where $ q$ is a complex number such that $ \vert q\vert<1$ and $ \chi$ is the nontrivial Dirichlet character modulo $ 4$. For odd $ s$, these expressions are connected with the automorphic world, in particular with Eisenstein series of level $ 4$. From this, we derive through Nesterenko's work the transcendance of the numbers $ \beta_q(2s+1)$ for $ q$ algebraic such that $ 0<\vert q\vert<1$. Our main result concerns the nature of the numbers $ \beta_q(2s)$: we give a lower bound for the dimension of the vector space over $ \mathbb{Q}$ spanned by $ 1,\beta_q(2),\beta_q(4),\dots,\beta_q(A)$, where $ 1/q\in\mathbb{Z}\setminus\{-1;1\}$ and $ A$ is an even integer. As consequences for $ 1/q\in\mathbb{Z}\setminus\{-1;1\}$, on the one hand there is an infinity of irrational numbers among $ \beta_q(2),\beta_q(4),\dots$, and on the other hand at least one of the numbers $ \beta_q(2),\beta_q(4),\dots, \beta_q(20)$ is irrational.


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Additional Information

Frédéric Jouhet
Affiliation: CNRS, UMR 5208 Institut Camille Jordan, Université de Lyon, Université Lyon I, Bâtiment du Doyen Jean Braconnier, 43, bd du 11 Novembre 1918, 69622 Villeurbanne Cedex, France
Email: jouhet@math.univ-lyon1.fr

Elie Mosaki
Affiliation: CNRS, UMR 5208 Institut Camille Jordan, Université de Lyon, Université Lyon I, Bâtiment du Doyen Jean Braconnier, 43, bd du 11 Novembre 1918, 69622 Villeurbanne Cedex, France
Email: mosaki@math.univ-lyon1.fr

DOI: http://dx.doi.org/10.1090/S0002-9947-2010-05236-5
PII: S 0002-9947(2010)05236-5
Keywords: $q$-analogues of the values of Dirichlet’s beta function at integers, modular forms, irrationality, basic hypergeometric series.
Received by editor(s): March 31, 2009
Received by editor(s) in revised form: July 20, 2009
Published electronically: October 15, 2010
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.