Character analogues of Ramanujan type integrals involving the Riemann -function
Abstract
A new class of integrals involving the product of -functions associated with primitive Dirichlet characters is considered. These integrals give rise to transformation formulas of the type F(z, α,)=F(-z, β,)=F(-z,α,)=F(z,β,), 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 M\"obius functions are derived.
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