Abstract
In this note, by the example of approximate calculation of \(\pi^{2}\) and Apéry’s constant the effect relating to the phenomenon of the so-called ‘‘curious’’ approximation are discussed. The mentioned effect was revealed in works of the 1980’s in the process of studying various classical series. For the remainder of the generalized harmonic series integral representations, complete asymptotic expansion and asymptotically exact two-sided estimates are given. These estimates make it possible to strictly explain the effect when considering particular cases.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
REFERENCES
A. B. Kostin and V. B. Sherstyukov, ‘‘Asymptotic behavior of remainders of special number series,’’ J. Math. Sci. 251, 814–839 (2020).
A. Yu. Popov, Two-Sided Estimates of the Sums of Function Values at Integer Points and Their Applications: A Practical Guide (Univ., Pereslavl’-Zalesskij, 2016) [in Russian].
J. M. Borwein, P. B. Borwein, and P. B. Dilcher, ‘‘Pi, Euler numbers and asymptotic expansions,’’ Am. Math. Mon. 96 (8), 681–687 (1989).
A. B. Kostin, V. B. Sherstyukov, and D. G. Tsvetkovich, ‘‘On the approximation of \(\pi^{2}\),’’ in Computer Mathematics Systems and their Applications, Proceedings of the 22nd International Conference (2021), pp. 261–264.
E. Ya. Riekstins, Asymptotic Expansions of Integrals (Zinatne, Riga, 1974), Vol. 1 [in Russian].
Author information
Authors and Affiliations
Corresponding authors
Additional information
(Submitted by F. G. Avkhadiev)
Rights and permissions
About this article
Cite this article
Kostin, A.B., Sherstyukov, V.B. & Tsvetkovich, D.G. Enveloping of Riemann’s Zeta Function Values and Curious Approximation. Lobachevskii J Math 43, 624–629 (2022). https://doi.org/10.1134/S1995080222060178
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S1995080222060178