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
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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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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DOI: https://doi.org/10.1007/s10958-015-2415-4