A modular relation involving a generalized digamma function and asymptotics of some integrals containing Ξ(t)

2022/11/16 by Dixit, Atul, Kumar, Rahul
#Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2211.08763

Abstract

A modular relation of the form F(α, w)=F(β, iw), where i=√(-1) and αβ=1, is obtained. It involves the generalized digamma function ψ_w(a) which was recently studied by the authors in their work on developing the theory of the generalized Hurwitz zeta function ζ_w(s, a). The limiting case w→0 of this modular relation is a famous result of Ramanujan on page 220 of the Lost Notebook. We also obtain asymptotic estimate of a general integral involving the Riemann function Ξ(t) as α→∞. Not only does it give the asymptotic estimate of the integral occurring in our modular relation as a corollary but also some known results.

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