  Let $X_1,X_2,...$ be independent variables, each having a normal distribution
with negative mean $-\beta<0$ and variance 1. We consider the partial sums
$S_n=X_1+...+X_n$, with $S_0=0$, and refer to the process $\{S_n:n\geq0\}$ as
the Gaussian random walk. We present explicit expressions for the mean and
variance of the maximum $M=\max\{S_n:n\geq0\}.$ These expressions are in terms
of Taylor series about $\beta=0$ 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 $\beta\downarrow0$.
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.
