Abstract
We provide a general theorem for evaluating trigonometric Dirichlet series of the form \(\sum _{n \geqslant 1} \frac{f (\pi n \tau )}{n^s}\), where f is an arbitrary product of the elementary trigonometric functions, \(\tau \) 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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Notes
In Marvin Knopp’s work, the role of the matrices T and S is usually reversed.
References
Berndt, B.C.: Dedekind sums and a paper of GH. Hardy. J. Lond. Math. Soc. 13(2), 129–137 (1976)
Berndt, B.C.: Modular transformations and generalizations of several formulae of Ramanujan. Rocky Mt. J. Math. 7(1), 147–190 (1977)
Berndt, B.C.: Analytic Eisenstein series, theta-functions, and series relations in the spirit of Ramanujan. J. Reine Angew. Math. 303(304), 332–365 (1978)
Berndt, B.C.: Ramanujan’s Notebooks Part II. Springer, New York (1989)
Berndt, B.C.: Ramanujan’s Notebooks Part V. Springer, New York (1998)
Berndt, B.C., Straub, A.: On a secant Dirichlet series and Eichler integrals of Eisenstein series. Preprint, 2014. arXiv:1406.2273
Charollois, P., Greenberg, M.: Rationality of secant zeta values. Ann. Sci. Math. Que. 38(1), 1–6 (2014)
Gun, S., Murty, M.R., Rath, P.: Transcendental values of certain Eichler integrals. Bull. Lond. Math. Soc. 43(5), 939–952 (2011)
Komori, Y., Matsumoto, K., Tsumura, H.: Barnes multiple zeta-functions, Ramanujan’s formula, and relevant series involving hyperbolic functions. J. Ramanujan Math. Soc. 28(1), 49–69 (2013)
Lagrange, J.L.: Solution d’un problème d’arithmétique. In: Serret, J.-A. (ed.) Oeuvres de Lagrange, vol. 1, pp. 671–731. Gauthier-Villars, Paris, 1867–1892
Lalín, M.N., Rodrigue, F., Rogers, M.D.: Secant zeta functions. J. Math. Anal. Appl. 409(1), 197–204 (2014)
Lenstra Jr., H.W.: Solving the Pell equation. Not. Am. Math. Soc. 49(2), 182–192 (2002)
Lewin, L.: Polylogarithms and Associated Functions. North Holland, New York (1981)
Paşol, V., Popa, A.A.: Modular forms and period polynomials. Proc. Lond. Math. Soc. 107(4), 713–743 (2013)
Rivoal, T.: On the convergence of Diophantine Dirichlet series. Proc. Edinb. Math. Soc. 55(2), 513–541 (2012)
Worley, R.T.: Estimating \(\alpha -p/q\). J. Aust. Math. Soc. Ser. A 31(2), 202–206 (1981)
Zagier, D.: Elliptic modular forms and their applications. The 1-2-3 of Modular Forms. Springer, Berlin (2008)
Acknowledgments
I thank Florian Luca for sharing his insight into his proof [11, Theorem 1] of the convergence of the secant Dirichlet series, and I am grateful to Bruce Berndt for helpful comments on an early version of this paper. I also thank the referee for suggestions improving the presentation.
Author information
Authors and Affiliations
Corresponding author
Additional information
Dedicated to the memory of Marvin Knopp.
Rights and permissions
About this article
Cite this article
Straub, A. Special values of trigonometric Dirichlet series and Eichler integrals. Ramanujan J 41, 269–285 (2016). https://doi.org/10.1007/s11139-015-9698-4
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11139-015-9698-4