On Lerch.s transcendent and the Gaussian random walk

Let X1,.X2,.. be independent variables, each having a normal distribution with negative mean ..<0 and variance 1. We consider the partial sums Sn=X1+.+Xn, with S0=0, and refer to the process {Sn:n.0} as the Gaussian random walk. We present explicit expressions for the mean and variance of the maximum M=max.{Sn:n.0}. These expressions are in terms of Taylor series about .=0 with coefficients that involve the Riemann zeta function. Our results extend Kingman.s first-order approximation [Proc. Symp. on Congestion Theory (1965) 137.169] of the mean for ..0. We build upon the work of Chang and Peres [Ann. Probab. 25 (1997) 787.802], and use Bateman.s formulas on Lerch.s transcendent and Euler.Maclaurin summation as key ingredients.