The Ramanujan Journal

, Volume 41, Issue 1–3, pp 269–285 | Cite as

Special values of trigonometric Dirichlet series and Eichler integrals



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.


Trigonometric Dirichlet series Eichler integrals 

Mathematics Subject Classification

Primary 11F11 33E20 Secondary 11L03 33B30 


  1. 1.
    Berndt, B.C.: Dedekind sums and a paper of GH. Hardy. J. Lond. Math. Soc. 13(2), 129–137 (1976)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Berndt, B.C.: Modular transformations and generalizations of several formulae of Ramanujan. Rocky Mt. J. Math. 7(1), 147–190 (1977)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    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)MathSciNetMATHGoogle Scholar
  4. 4.
    Berndt, B.C.: Ramanujan’s Notebooks Part II. Springer, New York (1989)MATHGoogle Scholar
  5. 5.
    Berndt, B.C.: Ramanujan’s Notebooks Part V. Springer, New York (1998)CrossRefMATHGoogle Scholar
  6. 6.
    Berndt, B.C., Straub, A.: On a secant Dirichlet series and Eichler integrals of Eisenstein series. Preprint, 2014. arXiv:1406.2273
  7. 7.
    Charollois, P., Greenberg, M.: Rationality of secant zeta values. Ann. Sci. Math. Que. 38(1), 1–6 (2014)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Gun, S., Murty, M.R., Rath, P.: Transcendental values of certain Eichler integrals. Bull. Lond. Math. Soc. 43(5), 939–952 (2011)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    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)MathSciNetMATHGoogle Scholar
  10. 10.
    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, 1867–1892Google Scholar
  11. 11.
    Lalín, M.N., Rodrigue, F., Rogers, M.D.: Secant zeta functions. J. Math. Anal. Appl. 409(1), 197–204 (2014)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Lenstra Jr., H.W.: Solving the Pell equation. Not. Am. Math. Soc. 49(2), 182–192 (2002)Google Scholar
  13. 13.
    Lewin, L.: Polylogarithms and Associated Functions. North Holland, New York (1981)MATHGoogle Scholar
  14. 14.
    Paşol, V., Popa, A.A.: Modular forms and period polynomials. Proc. Lond. Math. Soc. 107(4), 713–743 (2013)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Rivoal, T.: On the convergence of Diophantine Dirichlet series. Proc. Edinb. Math. Soc. 55(2), 513–541 (2012)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Worley, R.T.: Estimating \(\alpha -p/q\). J. Aust. Math. Soc. Ser. A 31(2), 202–206 (1981)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Zagier, D.: Elliptic modular forms and their applications. The 1-2-3 of Modular Forms. Springer, Berlin (2008)Google Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of Illinois at Urbana-ChampaignChampaignUSA

Personalised recommendations