Character analogues of Ramanujan type integrals involving the Riemann Ξ-function

2011/02/14 by Atul Dixit, Dixit, Atul · 1 citation
Mathematics · #11M06 #11M35 #Advanced Mathematical Identities #Analytic Number Theory Research #FOS: Mathematics #Mathematical functions and polynomials #Number Theory (math.NT) #math.NT #msc:11M06 #msc:11M35

paper · pdf · doi:10.48550/arxiv.1102.2680

28 pages

arxiv created 2011/02/14 · openalex publication_date 2011/02/14 · arxiv updated 2011/02/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

A new class of integrals involving the product of Ξ-functions associated with primitive Dirichlet characters is considered. These integrals give rise to transformation formulas of the type F(z, α,χ)=F(-z, β,χ)=F(-z,α,χ)=F(z,β,χ), where αβ=1. New character analogues of transformation formulas of Guinand and Koshliakov as well as those of a formula of Ramanujan and its recent generalization are shown as particular examples. Finally, character analogues of a conjecture of Ramanujan, Hardy and Littlewood involving infinite series of Möbius functions are derived.

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