The Riesz means, or sometimes typical means, were introduced by M. Riesz and have been studied in connection with summability of Fourier series and of Dirichlet series [8] and [11]. In number-theoretic context, it is the Riesz sum rather than the Riesz mean that has been extensively studied. The Riesz sums appear as long as there appears the G-function. Cf. Remark 1 and [14]. As is shown below, the Riesz sum corresponds to integration while Landau's differencing is an analogue of differentiation. This integration-differentiation aspect has been the driving force of many researches on number-theoretic asymptotic formulas. Ingham's decent treatment [13] of the prime number theorem is one typical example. We state some efficient theorems that give asymptotic formulas for the sums of coefficients of the generating Dirichlet series not necessarily satisfying the functional equation.
Published in |
Pure and Applied Mathematics Journal (Volume 4, Issue 5-1)
This article belongs to the Special Issue Mathematical Aspects of Engineering Disciplines |
DOI | 10.11648/j.pamj.s.2015040501.13 |
Page(s) | 15-19 |
Creative Commons |
This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited. |
Copyright |
Copyright © The Author(s), 2015. Published by Science Publishing Group |
Riesz Sum, Riesz Mean, Dirichlet Series, Asymptotic Formula
[1] | Bruce C. Berndt and S. Kim, Logarithmic means and double series of Bessel functions, preprint 2014. |
[2] | Bruce C. Berndt and S. Kim, Identities for logarithmic means: A survey, preprint 2014. |
[3] | B. C. Berndt and M. I. Knopp, Hecke's theory of modular forms and Dirichlet series, World Sci., Singapore etc., 2008. |
[4] | W. E. Briggs, Some Abelian results for Dirichlet series, Mathematika 9 (1962), 49-53. |
[5] | R. G. Buschman, Asymptotic expressions for Pnaf(n) logr n, Pacic J. Math. 9 (1959), 9-12. |
[6] | K. Chakraborty, S. Kanemitsu and H. Tsukada, Vistas of special functionsII, World Scienti_c, London Singapore New Jersey, 2009. |
[7] | K. Chandrasekharan, Arithmetical functions, Springer Verl., Berlin-Heidelberg-New York 1970. |
[8] | K. Chandrasekharan and S. Minakshisundaram, Typical means, Oxford UP, Oxford 1952. |
[9] | K. Chandrasekharan and Raghavan Narasimhan, Functional equations with multiple gamma factors and the average order of arithmetical functions, Ann. of Math. (2) 76 (1962), 93-136. |
[10] | H. Davenport, Multiplicative number theory, 1st ed. Markham, Chicago 1967, 2nd ed. Springer, New York etc. 1980. |
[11] | G. H. Hardy and M. Riesz, The general theory of Dirichlet's series, CUP. Cambridge 1915; reprint, Hafner, New York 1972 . |
[12] | A. E. Ingham, The distribution of prime numbers. Cambridge Tracts Math. Math. Phys., No. 30 Stechert-Hafner, Inc., New York 1964. |
[13] | S. Kanemitsu, On the Riesz sums of some arithmetical functions, in p-adic L-functions and algebraic number thery Surikaiseki Kenkyusho Kokyuroku 411 (1981), 109-120. |
[14] | S. Kanemitsu and H. Tsukada, Contributions to the theory of zetafunctions modular relation supremacy, World Sci. London etc. 2014, 303 pp. |
[15] | E. Landau, Uber die Anzahl der Gitterpunkte in gewissen Bereichen (Zweite Mit.), Nachr. Ges. Wiss. Gottingen, Math.-Phys. Kl. (1915), 209-243=Collected Works Vol. 6, Thales Verl., Essen 1985, 308-342. |
[16] | S. Swetharanyam, Asymptotic expansions for certain type of sums involving the arithmetic functions in the theory of numbers, Math. Student (2) 28 (1960), 9-26. |
[17] | X.-H. Wang and N. -L. Wang, Modular-relation theoretic interpretation of M. Katsurada's results, preprint 2014. |
APA Style
Hailong Li, Qianli Yang. (2015). On the Riesz Sums in Number Theory. Pure and Applied Mathematics Journal, 4(5-1), 15-19. https://doi.org/10.11648/j.pamj.s.2015040501.13
ACS Style
Hailong Li; Qianli Yang. On the Riesz Sums in Number Theory. Pure Appl. Math. J. 2015, 4(5-1), 15-19. doi: 10.11648/j.pamj.s.2015040501.13
@article{10.11648/j.pamj.s.2015040501.13, author = {Hailong Li and Qianli Yang}, title = {On the Riesz Sums in Number Theory}, journal = {Pure and Applied Mathematics Journal}, volume = {4}, number = {5-1}, pages = {15-19}, doi = {10.11648/j.pamj.s.2015040501.13}, url = {https://doi.org/10.11648/j.pamj.s.2015040501.13}, eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.pamj.s.2015040501.13}, abstract = {The Riesz means, or sometimes typical means, were introduced by M. Riesz and have been studied in connection with summability of Fourier series and of Dirichlet series [8] and [11]. In number-theoretic context, it is the Riesz sum rather than the Riesz mean that has been extensively studied. The Riesz sums appear as long as there appears the G-function. Cf. Remark 1 and [14]. As is shown below, the Riesz sum corresponds to integration while Landau's differencing is an analogue of differentiation. This integration-differentiation aspect has been the driving force of many researches on number-theoretic asymptotic formulas. Ingham's decent treatment [13] of the prime number theorem is one typical example. We state some efficient theorems that give asymptotic formulas for the sums of coefficients of the generating Dirichlet series not necessarily satisfying the functional equation.}, year = {2015} }
TY - JOUR T1 - On the Riesz Sums in Number Theory AU - Hailong Li AU - Qianli Yang Y1 - 2015/07/17 PY - 2015 N1 - https://doi.org/10.11648/j.pamj.s.2015040501.13 DO - 10.11648/j.pamj.s.2015040501.13 T2 - Pure and Applied Mathematics Journal JF - Pure and Applied Mathematics Journal JO - Pure and Applied Mathematics Journal SP - 15 EP - 19 PB - Science Publishing Group SN - 2326-9812 UR - https://doi.org/10.11648/j.pamj.s.2015040501.13 AB - The Riesz means, or sometimes typical means, were introduced by M. Riesz and have been studied in connection with summability of Fourier series and of Dirichlet series [8] and [11]. In number-theoretic context, it is the Riesz sum rather than the Riesz mean that has been extensively studied. The Riesz sums appear as long as there appears the G-function. Cf. Remark 1 and [14]. As is shown below, the Riesz sum corresponds to integration while Landau's differencing is an analogue of differentiation. This integration-differentiation aspect has been the driving force of many researches on number-theoretic asymptotic formulas. Ingham's decent treatment [13] of the prime number theorem is one typical example. We state some efficient theorems that give asymptotic formulas for the sums of coefficients of the generating Dirichlet series not necessarily satisfying the functional equation. VL - 4 IS - 5-1 ER -