Special values of trigonometric Dirichlet series and Eichler integrals

We provide a general theorem for evaluating trigonometric Dirichlet series of the form ∑ n ⩾ 1 f ( π n τ ) n s , where f is an arbitrary product of the elementary trigonometric functions, τ a real quadratic irrationality and s an integer of the appropriate parity. This unifies a number of evaluation...

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Veröffentlicht in:The Ramanujan journal 2016-11, Vol.41 (1-3), p.269-285
1. Verfasser: Straub, Armin
Format: Artikel
Sprache:eng
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Zusammenfassung:We provide a general theorem for evaluating trigonometric Dirichlet series of the form ∑ n ⩾ 1 f ( π n τ ) n s , where f is an arbitrary product of the elementary trigonometric functions, τ a real quadratic irrationality and s an integer of the appropriate parity. This unifies a number of evaluations considered by many authors, including Lerch, Ramanujan and Berndt. Our approach is based on relating the series to combinations of derivatives of Eichler integrals and polylogarithms.
ISSN:1382-4090
1572-9303
DOI:10.1007/s11139-015-9698-4