On a secant Dirichlet series and Eichler integrals of Eisenstein series

Bruce C. Berndt, Armin Straub

Research output: Contribution to journalArticlepeer-review


We consider the secant Dirichlet series ψs(τ)=∑n=1∞sec(πnτ)ns, 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 ψ2m(r), with r> 0 rational, are rational multiples of π2 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.

Original languageEnglish (US)
Pages (from-to)827-852
Number of pages26
JournalMathematische Zeitschrift
Issue number3-4
StatePublished - Dec 1 2016


  • Eichler integrals
  • Eisenstein series
  • Ramanujan polynomials
  • Trigonometric Dirichlet series
  • Unimodular polynomials

ASJC Scopus subject areas

  • Mathematics(all)


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