Atul Dixit ; Rahul Kumar - A modular relation involving a generalized digamma function and asymptotics of some integrals containing Ξ(t)

hrj:10913 - Hardy-Ramanujan Journal, February 6, 2023, Volume 45 - 2022 - https://doi.org/10.46298/hrj.2023.10913
A modular relation involving a generalized digamma function and asymptotics of some integrals containing Ξ(t)Article

Authors: Atul Dixit 1; Rahul Kumar

  • 1 Indian Institute of Technology [Gandhinagar]

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.


Volume: Volume 45 - 2022
Published on: February 6, 2023
Imported on: February 6, 2023
Keywords: Ramanujan's Lost notebook,modular relation,Generalized Hurwitz zeta function,Generalized digamma function,asymptotic estimates,11M06, 41A60, 33E20.,[MATH]Mathematics [math]

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