CHARACTER ANALOGUES OF RAMANUJAN-TYPE INTEGRALS INVOLVING THE RIEMANN Ξ -FUNCTION

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, (chi) over bar) = F(-z, alpha, (chi) over bar) = F(z, beta,chi); where alpha beta = 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 Mobius functions are derived.