Osaka Journal of Mathematics

A generalization of functional limit theorems on the Riemann zeta process

Satoshi Takanobu

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Abstract

$\zeta(\cdot)$ being the Riemann zeta function, $\zeta_{\sigma}(t) := \frac{\zeta(\sigma + i t)}{\zeta(\sigma)}$ is, for $\sigma > 1$, a characteristic function of some infinitely divisible distribution $\mu_{\sigma}$. A process with time parameter $\sigma$ having $\mu_{\sigma}$ as its marginal at time $\sigma$ is called a Riemann zeta process. Ehm [2] has found a functional limit theorem on this process being a backwards Lévy process. In this paper, we replace $\zeta(\cdot)$ with a Dirichlet series $\eta(\cdot;a)$ generated by a nonnegative, completely multiplicative arithmetical function $a(\cdot)$ satisfying (3), (4) and (5) below, and derive the same type of functional limit theorem as Ehm on the process corresponding to $\eta(\cdot;a)$ and being a backwards Lévy process.

Article information

Source
Osaka J. Math., Volume 56, Number 4 (2019), 843-882.

Dates
First available in Project Euclid: 21 October 2019

Permanent link to this document
https://projecteuclid.org/euclid.ojm/1571623225

Mathematical Reviews number (MathSciNet)
MR4020640

Subjects
Primary: 60F17: Functional limit theorems; invariance principles
Secondary: 60G51: Processes with independent increments; Lévy processes 11M41: Other Dirichlet series and zeta functions {For local and global ground fields, see 11R42, 11R52, 11S40, 11S45; for algebro-geometric methods, see 14G10; see also 11E45, 11F66, 11F70, 11F72}

Citation

Takanobu, Satoshi. A generalization of functional limit theorems on the Riemann zeta process. Osaka J. Math. 56 (2019), no. 4, 843--882. https://projecteuclid.org/euclid.ojm/1571623225


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