Bohr–Jessen process and functional limit theorem

Abstract

The Bohr–Jessen limit theorem states that for each $\sigma >\frac{1}{2}$, there exists an asymptotic probability distribution of $log\zeta (\sigma +\sqrt{-1}\cdot )$. Here $\zeta (\cdot )$ is the Riemann zeta function, and $log\zeta (\cdot )$ is a primitive function of $\zeta \text{'}/\zeta$ on some simply connected domain of $\mathbb{C}$. In this paper, we generalize this limit theorem to a functional limit theorem and show a similar limit theorem for a continuous process $\{log\zeta (\sigma +\sqrt{-1}\cdot ){\}}_{\sigma >1/2}$, which we call the Bohr–Jessen functional limit theorem.