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

作　　者：ZHANG Yi-feng SHI San-ying 

机构地区：[1]School of Mathematics, Hefei University of Technology, Hefei 230009, China

出　　处：《Chinese Quarterly Journal of Mathematics》2017年第3期271-276,共6页数学季刊（英文版）

基　　金：Supported by National Natural Science Foundation of China（11201107）

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 ∫_1~X?~2(x)dx? ε{X^(3-6/m+3+ε)if m ≥ 3,X^(2+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.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） satisfying 0≤ x 〈 n. Define △（x） = ∑1≤n≤xτ（n） - α x, where a is the residue of the Dedekind zetafunction ^（s, K） at its simple pole s = 1. In this paper it is shown that ∫x1△2（x）dx≤ε{x3-6/m＋3＋εX2＋ε if m≥3,if m=2,for any non-Abelian polynomial f（x） and any ε 〉 O. This result constitutes an improvement upon that of Lii for the error terms on average.

关 键 词：Dedekind zeta function polynomial congruence mean square 

分 类 号：O156.4[理学—数学]