## 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:

 [1] Chandrasekharan, K.; Narasimhan, R., The approximate functional equation for a class of zeta-functions, Math. ann., 152, 30-64, (1963) · Zbl 0116.27001 [2] Drza, E.; Varukhina, L., On Mean values of some arithmeitc functions in number fields, Discrete math., 308, 4892-4899, (2008) · Zbl 1223.11114 [3] Heath-Brown, D.R., The twelfth power moment of the Riemann zeta function, Q. J. math., 29, 443-462, (1978) · Zbl 0394.10020 [4] Heath-Brown, D.R., The growth rate of the Dedekind zeta function on the critical line, Acta arith., 49, 323-339, (1988) · Zbl 0583.12011 [5] Ivić, A., The Riemann zeta function, (1985), John Wiley & Sons New York · Zbl 0583.10021 [6] Iwaniec, H.; Kowalski, E., Analytic number theory, Amer. math. soc. colloq. publ., vol. 53, (2004), Amer. Math. Soc. Providence · Zbl 1059.11001 [7] Landau, E., Einführung in die elementare and analytische theorie der algebraischen zahlen und der ideale, (1927), Teubner · JFM 53.0141.09 [8] Lü, Guangshi, On Mean values of some arithmetic functions in number fields, Acta. math. hungar, (2010) · Zbl 1249.11106 [9] Lü, Guangshi; Wang, YongHui, Note on the number of integral ideals in Galois extension, Sci. China ser. A, 53, (2010) · Zbl 1273.11160 [10] Meurman, T., The Mean twelfth power of Dirichlet L-functions on the critical line, Ann. acad. sci. fenn. math. diss., 52, 1-44, (1984) · Zbl 0538.10034 [11] Nowak, W.G., On the distribution of integral ideals in algebraic number theory fields, Math. nachr., 161, 59-74, (1993) · Zbl 0803.11061 [12] Pan, C.D.; Pan, C.B., Fundamentals of analytic number theory, (1991), Science Press Beijing, (in Chinese) [13] Swinnerton-Dyer, H.P.F., A brief guide to algebraic number theory, London math. soc. stud. texts, vol. 50, (2001), Cambridge University Press Cambridge · Zbl 0963.11001
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