Abstract. The Riemann zeta process is a stochastic process $\{Z(\sigma ), \sigma >1\} $ with independent increments and marginal distributions whose characteristic functions are proportional to the Riemann zeta function along vertical lines ${\msbm R}e s = \sigma $. We establish functional limit theorems for the zeta process and other related processes as arguments $\sigma $ approach the pole at $s=1$ of the zeta function (from above).

AMS Subject Classification (1991): 60G51, 60F17, 11N37

Keyword(s): Erdős--Kac theorem, functional limit theorem, geometric process, Riemann zeta function, zeta process

Received August 14, 2007, and in revised form March 13, 2008. (Registered under 6021/2009.)