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On Dirichlet L-functions with periodic coefficients and Eisenstein series

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Abstract

We prove a Lipschitz type summation formula with periodic coefficients. Using this formula, representations of the values at positive integers of Dirichlet L-functions with periodic coefficients are obtained in terms of Bernoulli numbers and certain sums involving essentially the discrete Fourier transform of the periodic function forming the coefficients. The non-vanishing of these L-functions at s = 1 are then investigated. There are additional applications to the Fourier expansions of Eisenstein series over congruence subgroups of \({SL_2(\mathbb{Z})}\) and derivatives of such Eisenstein series. Examples of a family of Eisenstein series with a high frequency of vanishing Fourier coefficients are given.

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  1. Department of Mathematics, Koç University, Sarıyer, Istanbul, 34450, Turkey

    Emre Alkan

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  1. Emre Alkan
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Correspondence to Emre Alkan.

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This article was dedicated to my parents.

E. Alkan is supported in part by the Tübitak Career Award and Distinguished Young Scholar Award, Tüba-Gebip of Turkish Academy of Sciences.

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Alkan, E. On Dirichlet L-functions with periodic coefficients and Eisenstein series. Monatsh Math 163, 249–280 (2011). https://doi.org/10.1007/s00605-010-0211-2

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