Transformation formulas for Lambert series

Authors:

Shigeru KANEMITSU, Takako KUZUMAKI

AMS Classification:

11F20, 11F66

Keywords:

Dedekin eta-function, functional equation, Lambert series, transformation formula

Abstract:

The aim of this paper is two-fold. In Section 1, we make a rather complete, annotated list of all existing proofs of the eta-transformation formula in which the Dedekind sums appear (save for those which are concerned with the partition function). This will help streamlining the situation and will clarify the historical aspect of the transformation formula. In Section 2, we shall derive the transformation formula, for certain Lambert series under the action of an arbitrary modular transformation (not only for the Spiegelung), from the functional equation for the corresponding zeta-functions. Our approach not only complements the Goldstein-de la Torre's results but also establishes the result corresponding to Apostol's work on Lambert series. It turns out that our zeta-functions can be expressed as the sum and the difference of the Hurwitz zeta-function, and so they satisfy the functional equation of Hecke's type. This fact elucidates the previous analysis of Goldstein-de la Torre.

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Kanemitsu&Kuzumaki-09.pdf

Vol. 4 (12), 2009