×

On moments of non-normal number fields. (English) Zbl 1505.11076

Let \(K\) be a number field and \(a_K(m)\) be the number of integral ideals of \(K\) of norm equal to \(m\). The \(l\)-th moment of the Dedekind zeta function is defined by \[ S_l(X)=\sum_{m\leq X} a_K^l(m). \] In the paper under review, the authors obtain an asymptotic formula for \(S_l(X)\) under some restrictions on \(K\) and the structure of the Galois group \(G\) of \(\bar{K}/\mathbb{Q}\) as follows \[ S_l(X)= XP_l(\log X) +O_\varepsilon(X^{\delta+\varepsilon}), \] where \(\delta<1\) and \(P_l\) is a polynomial whose degree can be explicitly calculated depending on \(G\) and \(K\).

MSC:

11F66 Langlands \(L\)-functions; one variable Dirichlet series and functional equations
11F30 Fourier coefficients of automorphic forms
11R42 Zeta functions and \(L\)-functions of number fields
20C30 Representations of finite symmetric groups
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Aggarwal, K., Weyl bound for GL(2) in t-aspect via a simple delta method, J. Number Theory, 208, 72-100 (2020) · Zbl 1446.11092
[2] Bourgain, J., Decoupling, exponential sums and the Riemann zeta function, J. Am. Math. Soc., 30, 205-224 (2017) · Zbl 1352.11065
[3] Chandrasekharan, K.; Good, A., On the number of integral ideals in Galois extensions, Monatshefte Math., 95, 2, 99-109 (1983) · Zbl 0498.12009
[4] Chandrasekharan, K.; Narasimhan, R., The approximate functional equation for a class of zeta-functions, Math. Ann., 152, 30-64 (1963) · Zbl 0116.27001
[5] Fomenko, O. M., Mean values associated with the Dedekind zeta function, J. Math. Sci. (N.Y.), 150, 3, 2115-2122 (2008)
[6] 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
[7] Iwaniec, H.; Kowalski, E., Analytic Number Theory, Amer. Math. Soc. Colloq. Publ., vol. 53 (2004), American Mathematical Society: American Mathematical Society Providence · Zbl 1059.11001
[8] Knapp, A. W., Introduction to the Langlands program, Proc. Symp. Pure Math., 61, 245-302 (1997) · Zbl 0903.11015
[9] Lü, G., Mean values connected with the Dedekind zeta-function of a non-normal cubic field, Open Math., 11, 2, 274-282 (2012) · Zbl 1292.11108
[10] Prasad, D.; Yogananda, C. S., A report on Artin’s holomorphy conjecture, (Bambah, R. P.; Dumir, V. C.; Hans-Gill, R. J., Number Theory (2000), Hindustan Book Agency: Hindustan Book Agency Gurgaon) · Zbl 0982.11063
[11] Söhne, P., An upper bound for Hecke zeta-functions with Grössencharacters, J. Number Theory, 66, 225-250 (1997) · Zbl 0885.11062
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.