The average behavior of the coefficients of Dedekind zeta function over square numbers

In this paper, we are interested in the average behavior of the coefficients of Dedekind zeta function over square numbers. In Galois fields of degree d   which is odd, when l⩾1l⩾1 is an integer, we have∑n⩽xa(n2)l=xPm(logx)+O(x1−3md+6+ε), where m=(l(d+1)/2)dl−1m=((d+1)/2)ldl−1, Pm(t)Pm(t) is a polynomial in t   of degree m−1m−1, and ε>0ε>0 is an arbitrarily small constant. By using our method, we also rectify the main terms of the k-dimensional divisor problem in some Galois fields over square numbers established by Deza and Varukhina (2008) [DV].