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**An introduction to probabilistic number theory.**
*(English)*
Zbl 1481.11001

Cambridge Studies in Advanced Mathematics 192. Cambridge: Cambridge University Press (ISBN 978-1-108-84096-5/hbk; 978-1-108-88822-6/ebook). xiv, 255 p. (2021).

The book under review is an important contribution to the literature devoted to probabilistic number theory and related fields.

The connection between number theory and probability theory was discovered around a century ago and since then it has been intensively studied, with various very interesting and unexpected results having been proven. During recent years though, much more light has been shed in the connections of probability theory with number theory, with researchers having obtained a deeper understanding and clearer view of this interplay. The book under review is mainly devoted to results that belong to number theory, but the author emphasizes the probabilistic aspects of the proofs in order to highlight the interplay between the two fields. An important aspect of the book is that it develops probabilistic number theory from scratch, summarizing essential results from number theory, mathematical analysis and probability. Additionally, focus is given to key enlightening examples. This makes the book really reader friendly.

More specifically, the book discusses topics such as the Chebyshev bias, universality of the Riemann zeta function, exponential sums, Kloosterman sums, and other.

The author masterfully composes a broad array of results and also features connections to other areas of mathematics, both of arithmetic flavor but also to areas such as ergodic theory, expander graphs, and other. Such a detailed, systematic and motivated discussion of probabilistic number theory, which is a highly active and evolving domain of research, constitutes a very valuable contribution to the literature.

The book is very well written – as expected by an author who has already contributed very widely used and important books – and certainly belongs to all libraries of universities and research institutes. It has all the attributes to make a classic textbook in this fascinating domain. It will certainly be very useful to a broad spectrum of graduate students as well as advanced researchers working on this vibrant area and its various interconnections with other fields.

The connection between number theory and probability theory was discovered around a century ago and since then it has been intensively studied, with various very interesting and unexpected results having been proven. During recent years though, much more light has been shed in the connections of probability theory with number theory, with researchers having obtained a deeper understanding and clearer view of this interplay. The book under review is mainly devoted to results that belong to number theory, but the author emphasizes the probabilistic aspects of the proofs in order to highlight the interplay between the two fields. An important aspect of the book is that it develops probabilistic number theory from scratch, summarizing essential results from number theory, mathematical analysis and probability. Additionally, focus is given to key enlightening examples. This makes the book really reader friendly.

More specifically, the book discusses topics such as the Chebyshev bias, universality of the Riemann zeta function, exponential sums, Kloosterman sums, and other.

The author masterfully composes a broad array of results and also features connections to other areas of mathematics, both of arithmetic flavor but also to areas such as ergodic theory, expander graphs, and other. Such a detailed, systematic and motivated discussion of probabilistic number theory, which is a highly active and evolving domain of research, constitutes a very valuable contribution to the literature.

The book is very well written – as expected by an author who has already contributed very widely used and important books – and certainly belongs to all libraries of universities and research institutes. It has all the attributes to make a classic textbook in this fascinating domain. It will certainly be very useful to a broad spectrum of graduate students as well as advanced researchers working on this vibrant area and its various interconnections with other fields.

Reviewer: Michael Th. Rassias (Zürich)

### MSC:

11-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to number theory |

11Kxx | Probabilistic theory: distribution modulo \(1\); metric theory of algorithms |

11Lxx | Exponential sums and character sums |

11M06 | \(\zeta (s)\) and \(L(s, \chi)\) |