Mean values related to the Dedekind zeta-function

Tang, Hengcai, and Wang, Youjun. "Mean values related to the Dedekind zeta-function." Czechoslovak Mathematical Journal 74.4 (2024): 1265-1274. <http://eudml.org/doc/299622>.

@article{Tang2024,
abstract = {Let $K/\mathbb \{Q\}$ be a nonnormal cubic extension which is given by an irreducible polynomial $g(x)=x^3+a x^2+b x+c$. Denote by $\zeta _\{K\}(s)$ the Dedekind zeta-function of the field $K$ and $a_K(n)$ the number of integral ideals in $K$ with norm $n$. In this note, by the higher integral mean values and subconvexity bound of automorphic $L$-functions, the second and third moment of $a_K(n)$ is considered, i.e., \[ \sum \_\{n\le x\}a\_K^2(n)=x P\_1(\log x)+O(x^\{5/7+\epsilon \}),\quad \sum \_\{n\le x\}a\_K^3(n)=x P\_4(\log x)+O(X^\{321/356+\epsilon \}), \] where $P_1(t)$, $P_4(t)$ are polynomials of degree 1, 4, respectively, $\epsilon >0$ is an arbitrarily small number.},
author = {Tang, Hengcai, Wang, Youjun},
journal = {Czechoslovak Mathematical Journal},
keywords = {cusp form; Dedekind zeta-function; $L$-function},
language = {eng},
number = {4},
pages = {1265-1274},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Mean values related to the Dedekind zeta-function},
url = {http://eudml.org/doc/299622},
volume = {74},
year = {2024},
}

TY - JOUR
AU - Tang, Hengcai
AU - Wang, Youjun
TI - Mean values related to the Dedekind zeta-function
JO - Czechoslovak Mathematical Journal
PY - 2024
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 74
IS - 4
SP - 1265
EP - 1274
AB - Let $K/\mathbb {Q}$ be a nonnormal cubic extension which is given by an irreducible polynomial $g(x)=x^3+a x^2+b x+c$. Denote by $\zeta _{K}(s)$ the Dedekind zeta-function of the field $K$ and $a_K(n)$ the number of integral ideals in $K$ with norm $n$. In this note, by the higher integral mean values and subconvexity bound of automorphic $L$-functions, the second and third moment of $a_K(n)$ is considered, i.e., \[ \sum _{n\le x}a_K^2(n)=x P_1(\log x)+O(x^{5/7+\epsilon }),\quad \sum _{n\le x}a_K^3(n)=x P_4(\log x)+O(X^{321/356+\epsilon }), \] where $P_1(t)$, $P_4(t)$ are polynomials of degree 1, 4, respectively, $\epsilon >0$ is an arbitrarily small number.
LA - eng
KW - cusp form; Dedekind zeta-function; $L$-function
UR - http://eudml.org/doc/299622
ER -