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

A modular relation of the form F(?,w)=F(?,iw), where i=??1and ??=1, is obtained. It involves the generalized digamma function ?w(a) which was recently studied by the authors in their work on developing the theory of the generalized Hurwitz zeta function ?w(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 ?(t) as ???. Not only does it give the asymptotic estimate of the integral occurring in our modular relation as a corollary but also some known results.