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Higher moment of coefficients of Dedekind zeta function. (English) Zbl 1466.11087

Let \(\zeta_K(s) = \sum_{n=1}^\infty\frac{a_K(n)}{n^s}\) be the Dedekind zeta-function of a non-normal cubic field \(K\) and let \[ S_{K,m}(x) = \sum_{n_1^2+n_2^2\le x}a_K^m(n_1^2+n_2^2). \]
The equality \[ S_{K,1}(x) = xP(\log x) +O_\varepsilon\left(x^{3/5+\varepsilon}\right), \] where \(P(X)\) is a quadratic polynomial, has been established by Z. Yang [Front. Math. China 12, No. 4, 981–992 (2017; Zbl 1427.11122)]. The author shows that for \(2\le m\le 8\) one has \[ S_{K,m}(x) = xP_m(\log x) +O_\varepsilon\left(x^{\theta_m+\varepsilon}\right), \] where \(P_m(X)\) is a polynomial of degree \(\eta_m\), the numbers \(\theta_m\) and \(\eta_m\) given explicitly. For example, one has \(\eta_2=1, \theta_2=51/59\) and \(\eta_8=609, \theta_8=18356/18359\).

MSC:

11R42 Zeta functions and \(L\)-functions of number fields
11E25 Sums of squares and representations by other particular quadratic forms
11F30 Fourier coefficients of automorphic forms
11N37 Asymptotic results on arithmetic functions
11R16 Cubic and quartic extensions

Citations:

Zbl 1427.11122
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References:

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