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 StraubEmail author


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.


Eichler integrals Eisenstein series Trigonometric Dirichlet series Unimodular polynomials Ramanujan polynomials 

Mathematics Subject Classification

Primary 11F11 33E20 Secondary 11L03 33B30 



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.


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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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