On a secant Dirichlet series and Eichler integrals of Eisenstein series

TL;DR

This paper studies the secant Dirichlet series, proves rationality of certain values, links them to Eichler integrals of Eisenstein series, and explores the roots of associated period polynomials across levels.

Contribution

It establishes the rationality of secant Dirichlet series values at rational points and connects these series to Eichler integrals of Eisenstein series of higher levels.

Findings

Values of $\, ext{psi}_{2m}(\, extsqrt{r})$ are rational multiples of $\, extpi^{2m}$ for rational $r$.

Secant Dirichlet series are Eichler integrals of odd weight Eisenstein series.

Period polynomials at higher levels have roots mostly on the unit circle, similar to the level 1 case.

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 Lal\'{\i}n, Rodrigue and Rogers. In particular, we show, as conjectured and partially proven by Lal\'{\i}n, 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.