Ramanujan Series for Epstein Zeta Functions

Abstract

In the spirit of Ramanujan, we derive exponentially fast convergent series for Epstein zeta functions $E^{{\Gamma }_0(N)}(z,s)$ on the Hecke congruence groups ${\Gamma }_0(N),N\in {\mathbb{Z}}_{>0}$, 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).

Minor changes. An addendum to arXiv:1312.6352v4. 15 pages

References in corpus (1)

• Kontsevich-Zagier Integrals for Automorphic Green's Functions. I

Cited by in corpus (2)

• Kontsevich-Zagier Integrals for Automorphic Green's Functions. II
• Two Definite Integrals Involving Products of Four Legendre Functions