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 ∫1^X?2（x）dx? 

ε{X^3-6/m+3+εif 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. 

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