Ramanujan series for Epstein zeta functions

2014/10/31 by Yajun Zhou
Mathematics · #Absolute convergence #Advanced Algebra and Geometry #Advanced Mathematical Identities #Analytic Number Theory Research #Arithmetic zeta function #Congruence (geometry) #Convergent series #Modular form #Ramanujan's sum #Riemann zeta function #Series (stratigraphy) #Series expansion #math.CA #math.NT

paper · pdf · doi:10.1007/s11139-015-9695-7

published as Ramanujan J (2016) 40:367-388 · Minor changes. An addendum to arXiv:1312.6352v4. 15 pages

arxiv created 2015/03/28 · openalex publication_date 2015/06/15 · arxiv updated 2016/04/29 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05

Abstract

In the spirit of Ramanujan, we derive exponentially fast convergent series for Epstein zeta functions E^{\varGamma₀(N)}(z,s) on the Hecke congruence groups \varGamma₀(N),N∈\mathbb Z_>0, where z is an arbitrary point in the upper half-plane \mathfrak H, and s∈\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).

Citations

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• Heegner points and derivatives ofL-series. II
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