Third power moments of the error terms corresponding to certain arithmetic functions

Abstract

For positive integersn, letd(l 1,M 1;l 2,M 2;n) denote the number of factorizationsn=n 1 n 2 where each of the factorsn∈ℕ belongs to a prescribed congruence classl i moduloM i (i=1,2). In this article an asymptotic result is derived for the third power moment of the error term in the formula for the Dirichlet summmatory function ofd(l 1,M 1;l 2,M 2;n). This extends a recent result of [17] for the classic “unrestricted” case ofd(n)=d(1,1;1,1; n). Moreover, the technique developed is applied to the analogous problem related to Fourier coefficients of cusp forms.