Advanced Studies in Pure Mathematics

Some highlights from the history of probabilistic number theory

Wolfgang Schwarz

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In this survey lecture it is intended to sketch some parts [chosen according to the author's interests] of the [early] history of Probabilistic Number Theory, beginning with Paul Turáns proof (1934) of the Hardy—Ramanujan result on the “normal order” of the additive function $\omega (n)$, the Erdős–Wintner Theorem, and the Erdős–Kac Theorem. Next, mean–value theorems for arithmetical functions, and the Kubilius model and its application to limit laws for additive functions will be described in short.

Subsuming applications of the theory of almost–periodic functions under the concept of "Probabilistic Number Theory", the problem of "uniformly–almost–even functions with prescribed values" will be sketched, and the Knopfmacher – Schwarz – Spilker theory of integration of arithmetical functions will be sketched. Next, K.–H. Indlekofers elegant theory of integration of functions $\mathbb{N} \to \mathbb{C}$ of will be described.

Finally, it is tried to scratch the surface of the topic "universality", where important contributions came from the university of Vilnius.

Article information

Probability and Number Theory — Kanazawa 2005, S. Akiyama, K. Matsumoto, L. Murata and H. Sugita, eds. (Tokyo: Mathematical Society of Japan, 2007), 367-419

Received: 30 January 2006
Revised: 12 September 2006
First available in Project Euclid: 27 January 2019

Permanent link to this document euclid.aspm/1548550907

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11-02: Research exposition (monographs, survey articles)
Secondary: 11K65: Arithmetic functions [See also 11Nxx] 11N37: Asymptotic results on arithmetic functions 11M35: Hurwitz and Lerch zeta functions 30-02: Research exposition (monographs, survey articles) 30D99: None of the above, but in this section


Schwarz, Wolfgang. Some highlights from the history of probabilistic number theory. Probability and Number Theory — Kanazawa 2005, 367--419, Mathematical Society of Japan, Tokyo, Japan, 2007. doi:10.2969/aspm/04910367.

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