Brownian behaviour of the Riemann zeta function around the critical line

We establish a Brownian extension to Selberg's central limit theorem for the Riemann zeta function. This implies various limiting distributions for ζ, including an analogue of the reflection principle for the maximum of the Brownian motion: as T diverges, for any $u>0$ we have \[ 1/T· { meas}\{0≤ t≤ T:maxσ≥ 1/2log|ζ(σ+i t)|≥ u √1/2log log T \}→ 2 ∫_u∞ {e-x^2/2}{√2π}d x. \]