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

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 Möbius functions are derived.

28 pages