The average behavior of the coefficients of Dedekind zeta function over square numbers
Let \(K\) be an algebraic number field of degree \(d\) over rational field \(\mathbb{Q}\), \(\zeta_k(s)\) be its Dedekind zeta function and \(a(n)\) denotes the number of integral ideals in \(K\) with norm \(n\). Then \[ \zeta_k(s)=\sum\limits_{n=1}^{\infty}\frac{a(n)}{n^s}\,, \quad s=\sigma+it, \,\sigma>1\,. \] In this paper authors give the asymptotic behaviour of \(a(n)\) in Galois extension over \(\mathbb{Q}\) of odd degree \(d\) and in quadratic field. Using the same method they study \(k\)-dimensional divisor problem in the above fields. The main results are: \[ \sum\limits_{n\leq x}a(n^2)^l=xP_m(\log x)+R, \] where \(l\geq 1\), \(P_m(t)\) is polynomial in \(t\) of degree \(m-1\), \(\varepsilon>0\) is arbitrary small, \(m=\newline ((d+1)/2)^ld^{l-1}\), \(R=O(x^{1-\frac{1}{md+6}+\varepsilon})\) in the first case and \(m=M+1,\,M=(3^l-1)/2\), \(R=O(x^{1-\frac{3}{2m+2}+\varepsilon})\) in the second case; and \[ \sum\limits_{n\leq x}\tau_k^K(n^2)=xP_m(\log x)+R, \] where \(k\geq 2\) is an integer, \(m=(k^2d+k)/2\) in the first case and \(m=k^2+k\) in the second case. The above result rectify the main term of \textit{E. Deza} and \textit{L. Varukhina} [``On mean values of some arithmetic functions in number fields'', Discrete Math. 308, No. 21, 4892--4899 (2008; Zbl 1223.11114)] results and generalize the results in some Galois fields.
scientific article; zbMATH DE number 5931165
scientific article