Ramanujan Series for Epstein Zeta Functions

TL;DR

This paper develops rapidly converging series for Epstein zeta functions associated with Hecke groups, utilizing Ramanujan-inspired methods and Eichler integrals, extending recent results in modular form theory.

Contribution

It introduces exponentially fast convergent Ramanujan series for Epstein zeta functions on Hecke groups, connecting them with Eichler integrals and generalizing recent findings.

Findings

Derived new Ramanujan series for Epstein zeta functions

Reformulated series as Eichler integral-based modular form integrations

Extended recent results by Wan and Zucker

Abstract

In the spirit of Ramanujan, we derive exponentially fast convergent series for Epstein zeta functions $ E^{\varGamma_0(N)}(z,s)$ on the Hecke congruence groups $ \varGamma_0(N),N\in\mathbb Z_{>0}$, where $z$ is an arbitrary point in the upper half-plane $ \mathfrak H$, and $s\in\mathbb Z_{>1}$. These Ramanujan series can be reformulated as integrations of modular forms, in the framework of Eichler integrals. Particular cases of these Eichler integrals recover part of the recent results reported by Wan and Zucker (arXiv:1410.7081v1).