On the Dedekind zeta function.~II

O. M. Fomenko

Zapiski Nauchnykh Seminarov POMI, Analytical theory of numbers and theory of functions. Part 29, Tome 429 (2014), pp. 178-192

Résumé

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 $\Delta(x,K_n)$.