Osaka Journal of Mathematics

A generalization of functional limit theorems on the Riemann zeta process

Satoshi Takanobu

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$\zeta(\cdot)$ 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

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

First available in Project Euclid: 21 October 2019

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Mathematical Reviews number (MathSciNet)

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}


Takanobu, Satoshi. A generalization of functional limit theorems on the Riemann zeta process. Osaka J. Math. 56 (2019), no. 4, 843--882.

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  • P. Billingsley: Convergence of Probability Measures, 2nd edition, Wiley-Interscience Publ., 1999.
  • W. Ehm: A Riemann zeta stochastic process, C. R. Acad. Sci. Paris 345 (2007), 279–282.
  • W. Ehm: Functional limits of zeta type processes, Acta Sci. Math. (Szeged) 74 (2008), 381–398.
  • W. Feller: An Introduction to Probability Theory and Its Applications, vol. II, 2nd edition, John Wiley & Sons, 1971.
  • G.H. Hardy and E.M. Wright: An Introduction to the Theory of Numbers, 6th edition, Oxford Univ. Press, 2008.
  • K. Itô: Stochastic Processes, Springer, 2004.
  • Y. Kasahara and S. Watanabe: Limit theorems for point processes and their functionals, J. Math. Soc. Japan, 38 (1986), 543–574.
  • T. Lindvall: Weak convergence of probability measures and random functions in the function space $D[0,\infty)$, J. Appl. Prob., 10 (1973), 109–121.
  • W. Rudin: Principles of Mathematical Analysis, 3rd edition, McGraw-Hill, 1976.
  • G. Tenenbaum: Introduction to Analytic and Probabilistic Number Theory, Cambridge Univ. Press, 1995.