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

Atul Dixit

Vol. 255 (2012), No. 2, 317–348

DOI: 10.2140/pjm.2012.255.317

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

where αβ = 1. New character analogues of the Ramanujan–Guinand formula, the Koshliakov’s formula, and a transformation formula of Ramanujan, as well as its recent generalization, are shown as particular examples. Finally, character analogues of a conjecture of Ramanujan, and Hardy and Littlewood involving infinite series of Möbius functions are derived.

Keywords

Dirichlet character, Dirichlet L-function, modified Bessel function, Möbius function, Mellin transform, Ramanujan, Hardy, Littlewood, Koshliakov, Guinand

Mathematical Subject Classification 2010

Primary: 11M06

Secondary: 11M35

Milestones

Received: 13 February 2011

Accepted: 25 July 2011

Published: 10 April 2012