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

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<inf>w</inf>(a) which was recently studied by the authors in their work on developing the theory of the generalized Hurwitz zeta function ζ<inf>w</inf>(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.