## Kyoto Journal of Mathematics

### Bohr–Jessen process and functional limit theorem

Satoshi Takanobu

#### Abstract

The Bohr–Jessen limit theorem states that for each $\sigma\gt \frac{1}{2}$, 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 ${\zeta'}/{\zeta}$ 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 $\{\log\zeta(\sigma+\sqrt{-1}\cdot)\}_{\sigma\gt {1}/{2}}$, which we call the Bohr–Jessen functional limit theorem.

#### Article information

Source
Kyoto J. Math., Volume 54, Number 2 (2014), 401-426.

Dates
First available in Project Euclid: 2 June 2014

Permanent link to this document
https://projecteuclid.org/euclid.kjm/1401741284

Digital Object Identifier
doi:10.1215/21562261-2642440

Mathematical Reviews number (MathSciNet)
MR3215573

Zentralblatt MATH identifier
1302.60060

Subjects
Primary: 60F17: Functional limit theorems; invariance principles
Secondary: 11M06: $\zeta (s)$ and $L(s, \chi)$

#### Citation

Takanobu, Satoshi. Bohr–Jessen process and functional limit theorem. Kyoto J. Math. 54 (2014), no. 2, 401--426. doi:10.1215/21562261-2642440. https://projecteuclid.org/euclid.kjm/1401741284

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