On generalized Eisenstein series and Ramanujan's formula for periodic zeta-functions

2016/02/22 by M. Cihat Dağlıand Mümün Can
Mathematics · #math.NT #msc:11M36 #msc:11M41 #msc:11F20 #msc:11B68

paper · pdf · doi:10.1007/s00605-017-1020-7

published as Monatsh Math 184 (2017) 77-103 · arXiv admin note: text overlap with arXiv:1506.01809

arxiv created 2016/02/22 · arxiv updated 2017/09/21

Abstract

In this paper, transformation formulas for a large class of Eisenstein series defined by G(z,s;A_α,B_β;r₁,r₂)=∑_m,n=-∞^∞ \hspace-0.19in^^′\fracf(αm)f^∗(βn) ((m+r₁)z+n+r₂)^s, Re(s)>2, Im(z)>0 are investigated for s=1-r, r∈ℕ. Here \ f(n)\ and \ f^∗(n)\, -∞<n<∞ are sequences of complex numbers with period k>0, and A_α=\ f(αn)\ and B_β=\ f^∗(βn)\, α,β∈ℤ. Appearing in the transformation formulas are generalizations of Dedekind sums involving the periodic Bernoulli function. Reciprocity law is proved for periodic Apostol-Dedekind sum outside of the context of the transformation formulas. Furthermore, transformation formulas are presented for G(z,s;A_α,I;r₁,r₂) and G(z,s;I,A_α;r₁,r₂), where I=\ 1\. As an application of these formulas, analogues of Ramanujan's formula for periodic zeta-functions are derived.

Citations

• Periodic analogues of Dedekind sums and transformation formulas of Eisenstein series