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 |
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Format: | Artikel |
Sprache: | eng |
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Online-Zugang: | Volltext |
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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. |
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ISSN: | 1382-4090 1572-9303 |
DOI: | 10.1007/s11139-015-9698-4 |