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

Atul Dixit

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arXiv · 1102.2680

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

Atul Dixit

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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 $F(z, α,χ)=F(-z, β,\barχ)=F(-z,α,\barχ)=F(z,β,χ)$, 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 Möbius functions are derived.

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Atul Dixit. 2011-02-14. Character analogues of Ramanujan type integrals involving the Riemann $Ξ$-function. https://arxiv.org/abs/1102.2680

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