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(α,w)=F(β,iw), where i=1 and αβ=1, is obtained. It involves the generalized digamma function ψw(a) which was recently studied by the authors in their work on developing the theory of the generalized Hurwitz zeta function ζw(s,a). The limiting case w0 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.


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]

Consultation statistics

This page has been seen 499 times.
This article's PDF has been downloaded 331 times.