Functional limits of zeta type processes

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