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

2008/11/26 by Jouhet, Frederic, Mosaki, Elie
#Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.0811.4287

Abstract

small In this paper, we define q-analogues of Dirichlet's beta function at positive integers, which can be written as β_q(s)=∑_k≥1∑_d|kχ(k/d)d^s-1q^k for s∈\N^*, where q is a complex number such that |q|<1 and χ 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 β_q(2s+1) for q algebraic such that 0

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