caa a22 4500 605522545 CHVBK 20210128100745.0 cr unu---uuuuu 210128e20150601xx s 000 0 eng 10.1007/s10958-015-2415-4 doi (NATIONALLICENCE)springer-10.1007/s10958-015-2415-4 Fomenko O. St. Petersburg Department of the Steklov Mathematical Institute, St. Petersburg, Russia aut On the Dedekind Zeta Function. II [Elektronische Daten] [O. Fomenko] 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 = ℚ − 1 m 4 $$ {K}_8=\mathbb{Q}\left(\sqrt{-1},\sqrt[4]{m}\right) $$ , K 8 = ℚ ε m 4 $$ {K}_8=\mathbb{Q}\left(\sqrt[4]{\varepsilon_m}\right) $$ , and K 16 = ℚ − 1 ε m 4 $$ {K}_{16}=\mathbb{Q}\left(\sqrt{-1},\sqrt[4]{\varepsilon_m}\right) $$ , where m is a positive square-free integer and ε m denotes the fundamental unit of ℚ m $$ \mathbb{Q}\left(\sqrt{m}\right) $$ , the author proves that A x K n = Λ n x + Δ x K n x K n , Δ x K n ≪ x 1 − 3 n + 2 + ε . $$ \begin{array}{cc}\hfill A\left(x,{K}_n\right)={\Lambda}_nx+\Delta \left(x,{K}_n\right)\left(x,{K}_n\right),\hfill & \hfill \Delta \left(x,{K}_n\right)\ll {x}^{1-\frac{3}{n+2}+\varepsilon }.\hfill \end{array} $$ 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. Springer Science+Business Media New York, 2015 Journal of Mathematical Sciences Springer US; http://www.springer-ny.com 207/6(2015-06-01), 923-933 1072-3374 207:6<923 2015 207 10958 https://doi.org/10.1007/s10958-015-2415-4 text/html Onlinezugriff via DOI BK010053 XK010053 XK010000 Metadata rights reserved Springer special CC-BY-NC licence nationallicence 1 research-article jats NATIONALLICENCE NATIONALLICENCE NL-springer NATIONALLICENCE 856 40 https://doi.org/10.1007/s10958-015-2415-4 text/html Onlinezugriff via DOI NATIONALLICENCE 100 1- Fomenko O. St. Petersburg Department of the Steklov Mathematical Institute, St. Petersburg, Russia aut NATIONALLICENCE 773 0- Journal of Mathematical Sciences Springer US; http://www.springer-ny.com 207/6(2015-06-01), 923-933 1072-3374 207:6<923 2015 207 10958