Error term concerning number of subgroups of group {\text{ℤ}}_m\times {\text{ℤ}}_n with m^2+n^2\le x

Abstract

Let {\text{ℤ}}_m be the additive group of residue classes modulo m. Let s(m, n) denote the number of subgroups of the group {\text{ℤ}}_m\times {\text{ℤ}}_n, where m and n are arbitrary positive integers. For any x\ge 1, we consider the asymptotic behavior of D_s\left(x\right):=\sum m^2+n^2\le xS(M,n) and obtain an asymptotic formula by using the elementary method.

Keywords

Number of subgroups / asymptotic formula / error term / exponential sums

References

[1] Bourgain J, Watt N. Mean square of zeta function, circle problem and divisor problem revisited. arXiv: 1709.04340v1

[2] Fouvry E, Iwaniec H. Exponential sums with monomials. J Number Theory, 1989, 33: 311–333

[3] Hampejs M, Holighaus N, Tóth L, Wiesmeyr C. Representing and counting the subgroups of the group ℤm×ℤn. J Numbers, 2014, 2014: 491428

[4] Ivić A. The Riemann Zeta-Function: The Theory of the Riemann Zeta-Function with Applications. New York: Wiley, 1985

[5] Mertens F. Ueber einige asymptotische Gesetze der Zahlentheorie. J Reine Angew Math, 1874, 77: 289–338

[6] Nowak W G, Tóth L. On the average number of subgroups of the group ℤm×ℤn. Int J Number Theory, 2014, 10: 363–374

[7] Sui Y K, Liu D. On the error term concerning the number of subgroups of the groups ℤm×ℤn with mn ≤ x. J Number Theory, 2020, 216: 264–279

[8] Tóth L, Zhai W G. On the error term concerning the number of subgroups of the group ℤm×ℤn with m, n ≤ x. Acta Arith, 2018, 183(3): 285–299

[9] Voronoï G. Sur une fonction transcendante et ses applications à la sommation de quelques séries. Ann Sci Éc Norm Supér (3), 1904, 21: 207–267, 459–533

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