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On the Dedekind Zeta Function. II

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Let K n be a number field of degree n over ℚ. By A(x, K n ) denote the number of integral ideals of K n with norm ≤ x. For \( {K}_8=\mathbb{Q}\left(\sqrt{-1},\sqrt[4]{m}\right) \), \( {K}_8=\mathbb{Q}\left(\sqrt[4]{\varepsilon_m}\right) \), and \( {K}_{16}=\mathbb{Q}\left(\sqrt{-1},\sqrt[4]{\varepsilon_m}\right) \), where m is a positive square-free integer and ε m denotes the fundamental unit of \( \mathbb{Q}\left(\sqrt{m}\right) \), the author proves that

$$ \begin{array}{cc}\hfill A\left(x,{K}_n\right)={\Lambda}_nx+\Delta \left(x,{K}_n\right)\left(x,{K}_n\right),\hfill & \hfill \Delta \left(x,{K}_n\right)\ll {x}^{1-\frac{3}{n+2}+\varepsilon }.\hfill \end{array} $$

This improves earlier results of E. Landau (1917) and W. G. Nowak (Math. Nachr., 161 (1993), 59–74) for the special cases indicated.

Also the author treats Titchmarch’s phenomenon for ζK n (s) and large values of Δ(x, K n ). Bibliography: 26 titles.

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  1. St. Petersburg Department of the Steklov Mathematical Institute, St. Petersburg, Russia

    O. M. Fomenko

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Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 429, 2014, pp. 178–192.

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Fomenko, O.M. On the Dedekind Zeta Function. II. J Math Sci 207, 923–933 (2015). https://doi.org/10.1007/s10958-015-2415-4

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