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Article Dans Une Revue Transactions of the American Mathematical Society Année : 2011

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

Frederic Jouhet
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Résumé

small In this paper, we define $q$-analogues of Dirichlet's beta function at positive integers, which can be written as $\beta_q(s)=\sum_{k\geq1}\sum_{d|k}\chi(k/d)d^{s-1}q^k$ for $s\in\N^*$, where $q$ is a complex number such that $|q|<1$ and $\chi$ is the non trivial 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 $0<|q|<1$. 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 $\Q$ spanned by $1,\beta_q(2),\beta_q(4),\dots,\beta_q(A)$, where $1/q\in\Z\setminus\{-1;1\}$ and $A$ is an even integer. As consequences, for $1/q\in\Z\setminus\{-1;1\}$, 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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Dates et versions

hal-00341909 , version 1 (26-11-2008)

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Frederic Jouhet, Elie Mosaki. Diophantine properties for q-analogues of Dirichlet's beta function at positive integers. Transactions of the American Mathematical Society, 2011, 363, pp.1533-1554. ⟨hal-00341909⟩
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