Character analogues of Ramanujan type integrals involving the Riemann $\Xi$-function

TL;DR

This paper introduces a new class of integrals involving the Riemann $\\Xi$-function and Dirichlet characters, deriving transformation formulas and character analogues of classical results by Ramanujan, Guinand, and Koshliakov.

Contribution

It develops novel character analogues of transformation formulas and conjectures related to the Riemann $\Xi$-function and Dirichlet characters, expanding classical integral identities.

Findings

Derived new transformation formulas involving Dirichlet characters.

Established character analogues of Ramanujan, Hardy, and Littlewood conjectures.

Connected integrals with classical and recent results in analytic number theory.

Abstract

A new class of integrals involving the product of $\Xi$-functions associated with primitive Dirichlet characters is considered. These integrals give rise to transformation formulas of the type $F(z, \alpha,\chi)=F(-z, \beta,\bar{\chi})=F(-z,\alpha,\bar{\chi})=F(z,\beta,\chi)$, where $\alpha\beta=1$. 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 M\"{o}bius functions are derived.