On the asymptotics to all orders of the Riemann zeta function and of a two-parameter generalization of the Riemann zeta function. (English) Zbl 1498.11005

Memoirs of the American Mathematical Society 1351. Providence, RI: American Mathematical Society (AMS) (ISBN 978-1-4704-5098-4/pbk; 978-1-4704-7017-3/ebook). vii, 114 p. (2022).
In the research monograph under review, the authors establish some really interesting as well as strong results relevant to the asymptotics of the Riemann zeta function \(\zeta(s)\). More specifically, in this work they obtain several formulae for the large \(t\) asymptotics of \(\zeta(s)\), where \(s=\sigma+it\), \(0\leq \sigma \leq 1\), \(t>0\), which are valid to all orders. The generality of these results is also illustrated by the fact that the relevant classical results of Siegel are derived as special cases.
The authors apply the formulae they have established in order to obtain explicit representations for the sum \(\sum_a^b \frac{1}{n^s}\) for certain ranges of \(a, b\), as well as derive precise estimates relating the latter sum to the sum \(\sum_c^d \frac{1}{n^{s-1}}\), for certain ranges of \(a, b, c, d\).
Another important aspect of this work, is that the authors investigate a two-parameter generalization of the Riemann zeta function and they also obtain asymptotic formulae for this function as well.
The present monograph comprises of 10 Chapters, which are partitioned in 3 parts, namely: Part 1 Asymptotics to all orders of the Riemann zeta function, Part 2 Asymptotics to all orders of a two-parameter generalization of the Riemann zeta function, and Part 3 Representations for the basic sum. The monograph also contains 2 interesting appendices.
Overall, the monograph is very well written. The methods employed by the authors are demanding and highly technical, but they manage to make the content very clearly presented to the targeted audience, namely graduate students and advanced researchers.
The results presented in this monograph constitute important contributions in the domain of analysis and analytic number theory and specifically in the study of one of the most important functions investigated in analytic number theory, the Riemann zeta function.


11-02 Research exposition (monographs, survey articles) pertaining to number theory
11M06 \(\zeta (s)\) and \(L(s, \chi)\)
30E15 Asymptotic representations in the complex plane
33E20 Other functions defined by series and integrals


Full Text: DOI arXiv


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