Transformation formulas for Lambert series
Authors:


AMS Classification:

11F20, 11F66

Keywords:

Dedekin etafunction, functional equation, Lambert series, transformation formula

Abstract:

The aim of this paper is twofold. In Section 1, we make a rather complete, annotated list of all existing proofs of the etatransformation 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 zetafunctions. Our approach not only complements the Goldsteinde la Torre's results but also establishes the result corresponding to Apostol's work on Lambert series. It turns out that our zetafunctions can be expressed as the sum and the difference of the Hurwitz zetafunction, and so they satisfy the functional equation of Hecke's type. This fact elucidates the previous analysis of Goldsteinde la Torre.

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Vol. 4 (12), 2009
