×

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

Citations:

Zbl 1223.11114
PDF BibTeX XML Cite
Full Text: DOI

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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.