数学季刊 ›› 2017, Vol. 32 ›› Issue (3): 271-276.doi: 10.13371/j.cnki.chin.q.j.m.2017.03.006

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关于特定同余式方程解数的余项

  

  1. School of Mathematics, Hefei University of Technology
  • 收稿日期:2016-07-23 出版日期:2017-09-30 发布日期:2020-10-22
  • 作者简介:ZHANG Yi-feng(1990-), male, native of Luan, Anhui, a graduate student of Hefei University of Technology, engages in analytic number theory; SHI San-ying(1982-), female, native of Yugan, Jiangxi, an associate professor of Hefei University of Technology, Ph.D., engages in analytic number theory.
  • 基金资助:
    Supported by National Natural Science Foundation of China(11201107);

On the Error Term for the Number of Solutions of Certain Congruences

  1. School of Mathematics, Hefei University of Technology
  • Received:2016-07-23 Online:2017-09-30 Published:2020-10-22
  • About author:ZHANG Yi-feng(1990-), male, native of Luan, Anhui, a graduate student of Hefei University of Technology, engages in analytic number theory; SHI San-ying(1982-), female, native of Yugan, Jiangxi, an associate professor of Hefei University of Technology, Ph.D., engages in analytic number theory.
  • Supported by:

摘要:

Let f(x) be an irreducible polynomial of degree m ≥ 2 with integer coefficients,and let r(n) denote the number of solutions x of the congruence f(x) ≡ 0(mod n) satisfying0 ≤ x < n. Define ?(x) =Σ 1≤n≤x r(n)-αx, where α is the residue of the Dedekind zeta function ζ(s, K) at its simple pole s = 1. In this paper it is shown that ∫1X?2(x)dx? 

ε{X3-6/m+3+εif m ≥ 3,X2+ε if m = 2,for any non-Abelian polynomial f(x) and any ε > 0. This result constitutes an improvement upon that of Lü for the error terms on average. 

关键词: Dedekind zeta function, polynomial congruence, mean square

Abstract:

Let f(x) be an irreducible polynomial of degree m ≥ 2 with integer coefficients,and let r(n) denote the number of solutions x of the congruence f(x) ≡ 0(mod n) satisfying0 ≤ x < n. Define ?(x) =Σ 1≤n≤x r(n)-αx, where α is the residue of the Dedekind zeta function ζ(s, K) at its simple pole s = 1. In this paper it is shown that ∫1X?2(x)dx? 

ε{X3-6/m+3+εif m ≥ 3,X2+ε if m = 2,for any non-Abelian polynomial f(x) and any ε > 0. This result constitutes an improvement upon that of Lü for the error terms on average. 

Key words: Dedekind zeta function, polynomial congruence, mean square

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