Hardy-Ramanujan Journal |
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 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.