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New approach to probabilistic number theory – compactifications and integration. (English) Zbl 1228.11125
Akiyama, Shigeki (ed.) et al., Probability and number theory – Kanazawa 2005. Proceedings of the international conference on probability and number theory, Kanazawa, Japan, June 20–24, 2005. Tokyo: Mathematical Society of Japan (ISBN 978-4-931469-43-3/hbk). Advanced Studies in Pure Mathematics 49, 133-170 (2007).
Starting with a short historical introduction to the beginnings of probabilistic number theory (Hardy–Ramanujan’s result on $$\omega(n)$$, the Erdös–Kac result, the Kubilius model, the results of H. Delange, E. Wirsing, and G. Halász), the author gives his definition of uniformly summable functions in $$\mathcal L^1$$, where $\mathcal L^\alpha = \left\{f:\mathbb N \to\mathbb C, \|f\|_\alpha = \left\{\limsup_{x\to\infty} \frac1x\,\sum_{n\leq x} |f(n)|^\alpha\right\}^{\frac1\alpha} < \infty \right\}.$ $$f\in\mathcal L^1$$ is uniformly summable if $\lim_{K\to\infty}\sup_{N\geq 1} \frac1N\; \sum_{n\leq N,\, |f(n)|>K} |f(n)| =0.$ Then the author gives his generalizations of H. Delange’s result [Math. Z. 172, 255–271 (1980; Zbl 0416.10035)], of E. Wirsing’s result [J. Reine Angew. Math. 328, 116–127 (1981; Zbl 0455.10036)] and of G. Halász’s result [Period. Math. Hung. 17, 143–161 (1986; Zbl 0569.10024)], for uniformly summable functions (these functions assume “large” values only “rarely”). Next the author gives his results on limit-distributions for real-valued uniformly summable functions. After reviewing E. V. Novoselov’s theory of integration [see Izv. Akad. Nauk SSSR, Ser. Mat. 28, 307–364 (1964; Zbl 0213.33502)], the author describes the Stone–Čech compactification $$\beta\mathbb N$$ of $$\mathbb N$$, and some of its properties. The fundamental proposition is:
Given an algebra $$\mathcal A$$ in $$\mathbb N$$ and a finitely additive measure $$\delta: \mathcal A \to [0,\infty)$$ on $$\mathcal A$$, then the extension $$\bar{\delta}$$ of $$\delta$$ to $$\overline{\mathcal A}$$ (the extension of $$\mathcal A$$ to $$\beta\mathbb N$$) is $$\sigma$$-additive and can be (uniquely) extended to a measure on the minimal $$\sigma$$-algebra $$\sigma(\overline{\mathcal A})$$ over $$\overline{\mathcal A}$$.
Then the author sketches, how to construct candidates for $$\delta$$, using Toeplitz matrices, and he describes how the theory can be applied to uniformly summable functions.
The last section shows the vast richness of possible applications of the author’s theory: Using the Lindeberg–Levy martingale-theorem, the main result of the Kubilius-theory is obtained. Next, the author easily shows how to get the Erdös–Wintner three series theorem. A probabilistic proof of Euclid’s theorem (infinitude of the set of primes) is given. Using a lemma of Fürstenberg, Szemerédi’s theorem on arbitrary long arithmetic progressions in subsets of $$\mathbb N$$ with positive Banach density is proved. Finally the author obtains three-series characterizations of $$q$$-multiplicative functions in $$\mathcal L\ast$$ with $$\|f\|_1 > 0$$.
The referee thinks it should be possible to give more interesting applications of the author’s integration theory.
For the entire collection see [Zbl 1132.11001].

##### MSC:
 11K65 Arithmetic functions in probabilistic number theory 11N37 Asymptotic results on arithmetic functions 11N64 Other results on the distribution of values or the characterization of arithmetic functions