On The Error Term For The Number Of Solutions Of Certain Congruences

Let f(x)be an irreducible polynomial of degreem?2with integer coefficients,and let r(n)denote the number of solutions of the congruence f(x)? 0(mod n)satisfying 0?x<n.Define:?(x)= ?1?n?x r(n)-?x,where a is the residue of the Dedekind zeta function ?(s,K)at its simple pole s = 1.In this paper it is shown that?1x?(x)dx<<?X-6/m+3+? ifm?3m X2+? ifm=2 For any non-Abelian polynomial f(x)and any ?>0.This result constitutes an improvement upon that of Lii for the error terms on average.