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

Fokas, Athanassios S.; Lenells, Jonatan

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.

Reviewer: Michael Th. Rassias (Zürich)

MSC:

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

Keywords:

Riemann zeta function; asymptotics

Software:

DLMF

References:

[1]	Andersson, Johan, Mean value properties of the Hurwitz zeta-function, Math. Scand., 71, 2, 295-300 (1992) · Zbl 0815.11042
[2]	A. C. L. Ashton and A. S. Fokas, Relations among the Riemann zeta and Hurwitz zeta functions as well as their products, preprint.
[3]	Balanzario, Eugenio P.; S\'{a}nchez-Ortiz, Jorge, Riemann-Siegel integral formula for the Lerch zeta function, Math. Comp., 81, 280, 2319-2333 (2012) · Zbl 1292.11097
[4]	Balasubramanian, R., A note on Hurwitz’s zeta-function, Ann. Acad. Sci. Fenn. Ser. A I Math., 4, 1, 41-44 (1979) · Zbl 0405.10027
[5]	Berry, M. V., The Riemann-Siegel expansion for the zeta function: high orders and remainders, Proc. Roy. Soc. London Ser. A, 450, 1939, 439-462 (1995) · Zbl 0842.11030
[6]	Berry, M. V.; Keating, J. P., A new asymptotic representation for \(\zeta (\frac 12+it)\) and quantum spectral determinants, Proc. Roy. Soc. London Ser. A, 437, 1899, 151-173 (1992) · Zbl 0776.11048
[7]	Conrey, J. B.; Farmer, D. W.; Keating, J. P.; Rubinstein, M. O.; Snaith, N. C., Integral moments of \(L\)-functions, Proc. London Math. Soc. (3), 91, 1, 33-104 (2005) · Zbl 1075.11058
[8]	Edwards, H. M., Riemann’s zeta function, xiv+315 pp. (2001), Dover Publications, Inc., Mineola, NY · Zbl 1113.11303
[9]	Fernandez, Arran; Fokas, Athanassios S., Asymptotics to all orders of the Hurwitz zeta function, J. Math. Anal. Appl., 465, 1, 423-458 (2018) · Zbl 1444.11186
[10]	Garunkshtis, R.; Laurinchikas, A.; Steuding, I., An approximate functional equation for the Lerch zeta function, Mat. Zametki. Math. Notes, 74 74, 3-4, 469-476 (2003) · Zbl 1096.11034
[11]	Garunk\v{s}tis, R.; Laurin\v{c}ikas, A.; Steuding, J., On the mean square of Lerch zeta-functions, Arch. Math. (Basel), 80, 1, 47-60 (2003) · Zbl 1040.11064
[12]	Hardy, G. H.; Littlewood, J. E., The Approximate Functional Equations for zeta(s) and zeta2(s), Proc. London Math. Soc. (2), 29, 2, 81-97 (1929) · JFM 55.0203.05
[13]	K. Kalimeris and A. S. Fokas, Explicit asymptotics for certain single and double exponential sums, preprint, arXiv:1708.02868. · Zbl 1437.11120
[14]	Katsurada, Masanori; Matsumoto, Kohji, Discrete mean values of Hurwitz zeta-functions, Proc. Japan Acad. Ser. A Math. Sci., 69, 6, 164-169 (1993) · Zbl 0798.11034
[15]	Katsurada, Masanori; Matsumoto, Kohji, Explicit formulas and asymptotic expansions for certain mean square of Hurwitz zeta-functions. I, Math. Scand., 78, 2, 161-177 (1996) · Zbl 0871.11055
[16]	J. P. Keating, In Quantum chaos (eds. G. Casati, I. Guarneri, & U. Smilansky), Amsterdam: North-Holland, 1993, pp. 145-185.
[17]	Koksma, J. F.; Lekkerkerker, C. G., A mean-value theorem for \(\zeta (s,w)\), Nederl. Akad. Wetensch. Proc. Ser. A. {\bf 55} = Indagationes Math., 14, 446-452 (1952) · Zbl 0047.31501
[18]	Miyagawa, Takashi, Approximate functional equations for the Hurwitz and Lerch zeta-functions, Comment. Math. Univ. St. Pauli, 66, 1-2, 15-27 (2017) · Zbl 1414.11105
[19]	NIST Digital Library of Mathematical Functions, http://dlmf.nist.gov/, Release 1.0.20 of 2018-09-15. F. W. J. Olver, A. B. Olde Daalhuis, D. W. Lozier, B. I. Schneider, R. F. Boisvert, C. W. Clark, B. R. Miller, and B. V. Saunders, eds.
[20]	Olver, F. W. J., Asymptotics and special functions, xvi+572 pp. (1974), Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London · Zbl 0303.41035
[21]	Rane, V. V., On Hurwitz zeta function, Math. Ann., 264, 2, 147-151 (1983) · Zbl 0515.10037
[22]	Rane, Vivek V., A new approximate functional equation for Hurwitz zeta function for rational parameter, Proc. Indian Acad. Sci. Math. Sci., 107, 4, 377-385 (1997) · Zbl 0913.11036
[23]	B. Riemann, Uber die Anzahl der Primzahlen unter einer gegebenen Grosse, Monatsberichte der Koniglichen Preussischen Akademie der Wissenschaften zu Berlin (1859), 671-680.
[24]	C. L. Siegel, Uber Riemanns Nachlass zur analytischen Zahlentheorie, Quellen Studien zur Geschichte der Math. Astron. und Phys. Abt. B: Studien 2: 4580 (1932), reprinted in Gesammelte Abhandlungen, Vol. 1. Berlin: Springer-Verlag, 1966. · JFM 58.1037.07
[25]	Soundararajan, Kannan, Moments of the Riemann zeta function, Ann. of Math. (2), 170, 2, 981-993 (2009) · Zbl 1251.11058
[26]	Titchmarsh, E. C., The theory of the Riemann zeta-function, x+412 pp. (1986), The Clarendon Press, Oxford University Press, New York
[27]	Zhang, Wen Peng, On the Hurwitz zeta-function, Northeast. Math. J., 6, 3, 261-267 (1990) · Zbl 0732.11042
[28]	Zhang, Wen Peng, On the Hurwitz zeta-function, Illinois J. Math., 35, 4, 569-576 (1991) · Zbl 0713.11062
[29]	Zhang, Wen Peng, On the mean square value of the Hurwitz zeta-function, Illinois J. Math., 38, 1, 71-78 (1994) · Zbl 0788.11038