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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].

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