## The average behavior of the coefficients of Dedekind zeta function over square numbers.(English)Zbl 1261.11073

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 E. Deza and 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.

### MSC:

 11R42 Zeta functions and $$L$$-functions of number fields 11M41 Other Dirichlet series and zeta functions 11N37 Asymptotic results on arithmetic functions

Zbl 1223.11114
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### References:

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