Mathematische Zeitschrift

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

On a secant Dirichlet series and Eichler integrals of Eisenstein series

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 

References

  1. 1.
    Apostol, T.: Introduction to Analytic Number Theory. Springer, New York (1976)MATHGoogle Scholar
  2. 2.
    Ayoub, R.: Euler and the zeta function. Amer. Math. Monthly 81, 1067–1086 (1974)MathSciNetCrossRefMATHGoogle 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)MathSciNetCrossRefMATHGoogle 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)MathSciNetMATHGoogle Scholar
  6. 6.
    Berndt, B.C.: Dedekind sums and a paper of G. H. Hardy. J. Lond. Math. Soc. 13(2), 129–137 (1976)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Berndt, B.C.: Modular transformations and generalizations of several formulae of Ramanujan. Rocky Mt. J. Math. 7(1), 147–190 (1977)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Berndt, B.C.: Ramanujan’s Notebooks. Part II. Springer, New York (1989)MATHGoogle Scholar
  9. 9.
    Bol, G.: Invarianten linearer Differentialgleichungen. Abh. Math. Semin. Univ. Hambg. 16, 1–28 (1949)MathSciNetCrossRefMATHGoogle 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)MathSciNetMATHGoogle Scholar
  11. 11.
    Charollois, P., Greenberg, M.: Rationality of secant zeta values. Ann. Sci. Math. Quebec 38(1), 1–6 (2014)MathSciNetCrossRefMATHGoogle 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)MathSciNetCrossRefMATHGoogle 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)MathSciNetCrossRefMATHGoogle 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)MathSciNetCrossRefMATHGoogle 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)MathSciNetMATHGoogle Scholar
  17. 17.
    Katayama, K.: Ramanujan’s formulas for \(L\)-functions. J. Math. Soc. Jpn 26(2), 234–240 (1974)MathSciNetCrossRefMATHGoogle 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)MathSciNetMATHGoogle 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)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Lalín, M.N., Rodrigue, F., Rogers, M.D.: Secant zeta functions. J. Math. Anal. Appl. 409(1), 197–204 (2014)MathSciNetCrossRefMATHGoogle 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)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Miyake, T.: Modular Forms. Springer, Berlin (1989). Translated from the Japanese by Yoshitaka MaedaCrossRefMATHGoogle 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)MathSciNetMATHGoogle Scholar
  25. 25.
    Paşol, V., Popa, A.A.: Modular forms and period polynomials. Proc. Lond. Math. Soc. 107(4), 713–743 (2013)MathSciNetCrossRefMATHGoogle 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)MathSciNetCrossRefMATHGoogle 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)MathSciNetCrossRefMATHGoogle 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)MathSciNetCrossRefMATHGoogle 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

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