Advertisement

Mathematische Zeitschrift

, Volume 284, Issue 3–4, pp 827–852 | Cite as

On a secant Dirichlet series and Eichler integrals of Eisenstein series

  • Bruce C. Berndt
  • Armin Straub
Article

Abstract

We consider the secant Dirichlet series \(\psi _s (\tau ) = \sum _{n = 1}^{\infty } \frac{\sec (\pi n \tau )}{n^s}\), recently introduced and studied by Lalín, Rodrigue and Rogers. In particular, we show, as conjectured and partially proven by Lalín, Rodrigue and Rogers, that the values \(\psi _{2 m} (\sqrt{r})\), with \(r > 0\) rational, are rational multiples of \(\pi ^{2 m}\). We then put the properties of the secant Dirichlet series into context by showing that, for even s, they are Eichler integrals of odd weight Eisenstein series of level 4. This leads us to consider Eichler integrals of general Eisenstein series and to determine their period polynomials. In the level 1 case, these polynomials were recently shown by Murty, Smyth and Wang to have most of their roots on the unit circle. We provide evidence that this phenomenon extends to the higher level case. This observation complements recent results by Conrey, Farmer and Imamoglu as well as El-Guindy and Raji on zeros of period polynomials of Hecke eigenforms in the level 1 case. Finally, we briefly revisit results of a similar type in the works of Ramanujan.

Keywords

Eichler integrals Eisenstein series Trigonometric Dirichlet series Unimodular polynomials Ramanujan polynomials 

Mathematics Subject Classification

Primary 11F11 33E20 Secondary 11L03 33B30 

Notes

Acknowledgments

We thank Matilde Lalín, Francis Rodrigue and Mathew Rogers for sharing the preprint [21], which motivated the present work. We are very grateful to Alexandru Popa for making us aware of the recent paper [25] and for very helpful discussions, as well as to Bernd Kellner for comments on an earlier version of this paper. Finally, the second author would like to thank the Max-Planck-Institute for Mathematics in Bonn, where part of this work was completed, for providing wonderful working conditions.

References

  1. 1.
    Apostol, T.: Introduction to Analytic Number Theory. Springer, New York (1976)zbMATHGoogle Scholar
  2. 2.
    Ayoub, R.: Euler and the zeta function. Amer. Math. Monthly 81, 1067–1086 (1974)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Berndt, B.C.: Character transformation formulae similar to those for the Dedekind eta-function. In: Diamond, H. (ed.) Proceedings of Symposia in Pure Mathematics, vol. 24, pp. 9–30. American Mathematical Society, Providence (1973)Google Scholar
  4. 4.
    Berndt, B.C.: Character analogues of the Poisson and Euler–MacLaurin summation formulas with applications. J. Number Theory 7(4), 413–445 (1975)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Berndt, B.C.: On Eisenstein series with characters and the values of Dirichlet \(L\)-functions. Acta Arith. 28(3), 299–320 (1975)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Berndt, B.C.: Dedekind sums and a paper of G. H. Hardy. J. Lond. Math. Soc. 13(2), 129–137 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Berndt, B.C.: Modular transformations and generalizations of several formulae of Ramanujan. Rocky Mt. J. Math. 7(1), 147–190 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Berndt, B.C.: Ramanujan’s Notebooks. Part II. Springer, New York (1989)zbMATHGoogle Scholar
  9. 9.
    Bol, G.: Invarianten linearer Differentialgleichungen. Abh. Math. Semin. Univ. Hambg. 16, 1–28 (1949)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Conrey, J.B., Farmer, D.W., Imamoglu, Ö.: The nontrivial zeros of period polynomials of modular forms lie on the unit circle. Int. Math. Res. Not. 2013(20), 4758–4771 (2013)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Charollois, P., Greenberg, M.: Rationality of secant zeta values. Ann. Sci. Math. Quebec 38(1), 1–6 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Chowla, S.: Some infinite series, definite integrals and asymptotic expansions. J. Indian Math. Soc. 17, 261–288 (1927/28)Google Scholar
  13. 13.
    Cohn, A.: Über die Anzahl der Wurzeln einer algebraischen Gleichung in einem Kreise. Math. Z. 14(1), 110–148 (1922)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    El-Guindy, A., Raji, W.: Unimodularity of roots of period polynomials of Hecke eigenforms. Bull. Lond. Math. Soc. 46(3), 528–536 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Gun, S., Murty, M.R., Rath, P.: Transcendental values of certain Eichler integrals. Bull. Lond. Math. Soc. 43(5), 939–952 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Grosswald, E.: Die Werte der Riemannschen Zetafunktion an ungeraden Argumentstellen. Nachr. Akad. Wiss. Göttingen Math.-Phys. Kl. II 9–13, 1970 (1970)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Katayama, K.: Ramanujan’s formulas for \(L\)-functions. J. Math. Soc. Jpn 26(2), 234–240 (1974)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    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)Google Scholar
  19. 19.
    Lenstra Jr., H.W.: Solving the Pell equation. Not. Amer. Math. Soc. 49(2), 182–192 (2002)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Lalín, M.N., Rogers, M.D.: Variations of the Ramanujan polynomials and remarks on \(\zeta (2j+1)/\pi ^{2j+1}\). Funct. Approx. Comment. Math. 48(1), 91–111 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Lalín, M.N., Rodrigue, F., Rogers, M.D.: Secant zeta functions. J. Math. Anal. Appl. 409(1), 197–204 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Lalín, M.N., Smyth, C.J.: Unimodularity of zeros of self-inversive polynomials. Acta Math. Hung. 138(1–2), 85–101 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Miyake, T.: Modular Forms. Springer, Berlin (1989). Translated from the Japanese by Yoshitaka MaedaCrossRefzbMATHGoogle Scholar
  24. 24.
    Murty, M.R., Smyth, C.J., Wang, R.J.: Zeros of Ramanujan polynomials. J. Ramanujan Math. Soc. 26(1), 107–125 (2011)MathSciNetzbMATHGoogle Scholar
  25. 25.
    Paşol, V., Popa, A.A.: Modular forms and period polynomials. Proc. Lond. Math. Soc. 107(4), 713–743 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Razar, M.J.: Values of Dirichlet series at integers in the critical strip. In: Serre, J.-P., Zagier, D.B. (eds.) Modular Functions of One Variable VI. Number 627 in Lecture Notes in Mathematics, pp. 1–10. Springer, Berlin (1977)Google Scholar
  27. 27.
    Rivoal, T.: On the convergence of Diophantine Dirichlet series. Proc. Edinb. Math. Soc. 55(02), 513–541 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Weil, A.: Remarks on Hecke’s lemma and its use. In: Iyanaga, S. (ed.) Algebraic Number Theory: Papers Contributed for the Kyoto International Symposium, 1976, pp. 267–274. Japan Society for the Promotion of Science (1977)Google Scholar
  29. 29.
    Yoshida, M.: Hypergeometric functions, my love: modular interpretations of configuration spaces. Aspects of Mathematics, E32. Friedr. Vieweg & Sohn, Braunschweig (1997)Google Scholar
  30. 30.
    Zagier, D.: Periods of modular forms and Jacobi theta functions. Invent. Math. 104(1), 449–465 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Zucker, I.J., Robertson, M.M.: Some properties of Dirichlet \(L\)-series. J. Phys. A Math. Gen. 9(8), 1207–1214 (1976)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of Illinois at Urbana-ChampaignUrbanaUSA
  2. 2.Department of Mathematics and StatisticsUniversity of South AlabamaMobileUSA

Personalised recommendations