In pursuit of zeta-3. The world’s most mysterious unsolved math problem. (English) Zbl 1485.11002

Nahin, Paul J.

Princeton, NJ: Princeton University Press (ISBN 978-0-691-20607-3/hbk; 978-0-691-24764-9/pbk; 978-0-691-22759-7/ebook). xx, 320 p. (2021).

The book under review is devoted to a notoriously difficult and intriguing open problem, namely the solution of the so-called zeta-3 problem, that has baffled researchers for centuries.
Written by an acclaimed author, the present book very successfully communicates the mysteries of this problem, presenting basic background knowledge as well as a variety of related topics intended for a broad audience. The author has very carefully chosen the content, style and level of the book to both introduce and allure readers from a really broad spectrum. More specifically though, the book is primarily intended for mathematics enthusiasts with an understanding of basic calculus. However, I am sure that even advanced researchers in the domain will certainly appreciate the plethora of beautiful formulas and results presented within, as well as enjoy the very nice historical remarks carefully selected by the author and nicely presented as the content of the book progresses. As the author states, the general approach of the book is to keep the exposition intuitive, and he succeeds to do so. As mentioned above, apart from the zeta-3 problem the book also discusses other related fascinating topics, such as for example the Riemann Hypothesis – one of mathematics’ most important and celebrated open problems. No matter how complex the underlying notions are, the author manages to communicate the core ideas while attaining a reader-friendly conversational style of exposition.
The book comprises of four chapters entitled “Euler’s Problem”, “More wizard math and the zeta function”, “Periodic functions, Fourier series, and the zeta function”, “Euler sums, the harmonic series, and the zeta function”, as well as an Epilogue and five very interesting Appendices. Throughout the book, the author also invites the reader to challenge himself/herself with a number of proposed problems presented at the end of most sections, while providing also the solution to these questions at the end of the book after the Appendices.
Overall, for the reasons mentioned above, I am confident that this book will be both enjoyable and a rich source of useful as well as intriguing information to a wide range of readers.

Reviewer: Michael Th. Rassias (Zürich)

MSC:

11-01	Introductory exposition (textbooks, tutorial papers, etc.) pertaining to number theory
11M06	\(\zeta (s)\) and \(L(s, \chi)\)

Keywords:

Euler’s problem; zeta-3 problem; Riemann zeta function; harmonic series; Gamma function; periodic functions; Fourier series; Euler sums

Software:

Matlab