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Article Dans Une Revue Hardy-Ramanujan Journal Année : 2023

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

Résumé

A modular relation of the form $F(\alpha, w)=F(\beta, iw)$, where $i=\sqrt{-1}$ and $\alpha\beta=1$, is obtained. It involves the generalized digamma function $\psi_w(a)$ which was recently studied by the authors in their work on developing the theory of the generalized Hurwitz zeta function $\zeta_w(s, a)$. The limiting case $w\to0$ 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 $\Xi(t)$ as $\alpha\to\infty$. 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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Dates et versions

hal-03914268 , version 1 (28-12-2022)

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Citer

Atul Dixit, Rahul Kumar. A modular relation involving a generalized digamma function and asymptotics of some integrals containing Ξ(t). Hardy-Ramanujan Journal, 2023, Volume 45 - 2022, pp.140 -- 151. ⟨10.46298/hrj.2023.10913⟩. ⟨hal-03914268⟩
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