收稿日期: 2016-07-23
网络出版日期: 2020-10-22
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
Received date: 2016-07-23
Online published: 2020-10-22
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.
张义锋, 史三英 . 关于特定同余式方程解数的余项[J]. 数学季刊, 2017 , 32(3) : 271 -276 . DOI: 10.13371/j.cnki.chin.q.j.m.2017.03.006
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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