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On a secant Dirichlet series and Eichler integrals of Eisenstein series

On a secant Dirichlet series and Eichler integrals of Eisenstein series
Bruce C. Berndt, Armin Straub — Mathematische Zeitschrift — Volume 284, Number 3, 2016, Pages 827-852

Abstract

We consider, for even \(s\), 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 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.

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BibTeX

@article{secantsums-2016,
    author = {Bruce C. Berndt and Armin Straub},
    title = {On a secant {D}irichlet series and {E}ichler integrals of {E}isenstein series},
    journal = {Mathematische Zeitschrift},
    year = {2016},
    volume = {284},
    number = {3},
    pages = {827--852},
    doi = {10.1007/s00209-016-1675-0},
}