Rapidly convergent series representations of symmetric Tornheim double zeta functions

2021/03/31 by Nakamura, Takashi
#11M32 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2103.16873

Abstract

In the present paper, for s,t,u ∈ ℂ, we show rapidly (or globally) convergent series representations of the Tornheim double zeta function T(s,t,u) and (desingularized) symmetric Tornheim double zeta functions. As a corollary, we give a new a proof of known results on the values of T(s,s,s) at non-positive integers and the location of the poles of T(s,s,s). Furthermore, we prove that the function T(s,s,s) can not be written by a polynomial in the form of ∑_k=1^j c_k ∏_r=1^q ζ^d_kr (a_kr s + b_kr), where a_kr, b_kr, c_k ∈ ℂ and d_kr ∈ ℤ_{≥ 0}.

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