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Zap. Nauchn. Sem. POMI, 2014, Volume 429, Pages 178–192 (Mi znsl6074)  

This article is cited in 1 scientific paper (total in 1 paper)

On the Dedekind zeta function. II

O. M. Fomenko

St. Petersburg Department of V. A. Steklov Institute of Mathematics of the Russian Academy of Sciences, St. Petersburg, Russia

Abstract: Let $K_n$ be a number field of degree $n$ over $\mathbb Q$. Denote by $A(x,K_n)$ the number of integer ideals of $K_n$ with norm $\leq x$. For $K_8=\mathbb Q(\sqrt{-1},\root4\of m)$, $K_8=\mathbb Q(\root4\of{\varepsilon_m})$ and $K_{16}=\mathbb Q(\sqrt{-1},\root4\of{\varepsilon_m})$, where $m$ is a positive square free integer and $\varepsilon_m$ denotes the fundamental unit of $\mathbb Q(\sqrt m)$, the author proves that
$$ A(x,K_n)=\Lambda_nx+\Delta(x,K_n)(x,K_n),\quad\Delta(x,K_n)\ll x^{1-\frac3{n+2}+\varepsilon}. $$
This improves earlier results of E. Landau (1917) and W. G. Nowak (Math. Nachr. 161 (1993), 59–74) for the indicated special cases.
The author also treats Titchmarch's phenomenon for $\zeta_{K_n}(s)$ and large values of $\Delta(x,K_n)$.

Key words and phrases: Dedekind $\zeta$-function, extremal values.

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English version:
Journal of Mathematical Sciences (New York), 2015, 207:6, 923–933

UDC: 511.466+517.863
Received: 20.10.2014

Citation: O. M. Fomenko, “On the Dedekind zeta function. II”, Analytical theory of numbers and theory of functions. Part 29, Zap. Nauchn. Sem. POMI, 429, POMI, St. Petersburg, 2014, 178–192; J. Math. Sci. (N. Y.), 207:6 (2015), 923–933

Citation in format AMSBIB
\Bibitem{Fom14}
\by O.~M.~Fomenko
\paper On the Dedekind zeta function.~II
\inbook Analytical theory of numbers and theory of functions. Part~29
\serial Zap. Nauchn. Sem. POMI
\yr 2014
\vol 429
\pages 178--192
\publ POMI
\publaddr St.~Petersburg
\mathnet{http://mi.mathnet.ru/znsl6074}
\transl
\jour J. Math. Sci. (N. Y.)
\yr 2015
\vol 207
\issue 6
\pages 923--933
\crossref{https://doi.org/10.1007/s10958-015-2415-4}
\scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84949626578}


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    This publication is cited in the following articles:
    1. O. M. Fomenko, “On the mean square of the error term for Dedekind zeta functions”, J. Math. Sci. (N. Y.), 217:1 (2016), 125–137  mathnet  crossref  mathscinet
  • Записки научных семинаров ПОМИ
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