On certain sums of number theory

被引：14

作者：

Bordelles, Olivier ^{[1
]}

机构：

[1] 2 Allee Combe, F-43000 Aiguilhe, France

来源：

INTERNATIONAL JOURNAL OF NUMBER THEORY | 2022年 / 18卷 / 09期

关键词：

Dirichlet hyperbola principle; exponential sums of type I and II; Vaughan's identity; exponent pairs;

D O I：

10.1142/S1793042122501056

中图分类号：

O1 [数学];

学科分类号：

0701 ; 070101 ;

摘要：

We study sums of the shape Sigma(n <= x) f(<sic>x/n<sic>) where f is either the von Mangoldt function or the Dirichlet-Piltz divisor functions. We improve previous estimates when f = Lambda and f = tau, and provide new results when f = tau(r) with r >= 3, breaking the 1/2-barrier in each case. The functions f = mu(2), f = 2(omega) and f = omega are also investigated.

引用

收藏

页码：2053 / 2074

页数：22

相关论文

共 15 条

[1] Sums of two relatively prime cubes [J].

Baker, R. C. .

ACTA ARITHMETICA, 2007, 129 (02) :103-146

[2] On a sum involving the Euler function [J].

Bordelles, Olivier ;

Dai, Lixia ;

Heyman, Randell ;

Pan, Hao ;

Shparlinski, Igor E. .

JOURNAL OF NUMBER THEORY, 2019, 202 :278-297

[3] On a sum involving the number of distinct prime factors function related to the integer part function [J].

Bouderbala, Mihoub ;

Karras, Meselem .

NOTES ON NUMBER THEORY AND DISCRETE MATHEMATICS, 2020, 26 (04) :52-56

[4] DECOUPLING, EXPONENTIAL SUMS AND THE RIEMANN ZETA FUNCTION [J].

Bourgain, J. .

JOURNAL OF THE AMERICAN MATHEMATICAL SOCIETY, 2017, 30 (01) :205-224

[5]

Bourgain J., 2017, Mean square of zeta function, circle problem and divisor problem revisited

[6]

Davenport H., 2000, GRADUATE TEXTS MATH, V3rd

[7] Explicit bounds on exponential sums and the scarcity of squarefree binomial coefficients [J].

Granville, A ;

Ramare, O .

MATHEMATIKA, 1996, 43 (85) :73-107

[8]

Heath-Brown D.R, 2017, PROC MAT I STEKLOVA, V296, P95

[9] On a sum involving the divisor function [J].

Ma, Jing ;

Sun, Huayan .

PERIODICA MATHEMATICA HUNGARICA, 2021, 83 (02) :185-191

[10] On a sum involving the Mangoldt function [J].

Ma, Jing ;

Wu, Jie .

PERIODICA MATHEMATICA HUNGARICA, 2021, 83 (01) :39-48