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Keywords:
cusp form; Dedekind zeta-function; $L$-function
cusp form; Dedekind zeta-function; $L$-function
Summary:
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., $$ \sum _{n\leq x}a_K^2(n)=x P_1(\log x)+O(x^{5/7+\epsilon }),\quad \sum _{n\leq x}a_K^3(n)=x P_4(\log x)+O(X^{321/356+\epsilon }), $$ where $P_1(t)$, $P_4(t)$ are polynomials of degree 1, 4, respectively, $\epsilon >0$ is an arbitrarily small number.
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