The asymptotic distribution of coefficients of the Dedekind zeta-function over a sparse sequence
Volume 175 / 2024
Colloquium Mathematicum 175 (2024), 13-30
MSC: Primary 11F30; Secondary 11N37, 11R42
DOI: 10.4064/cm9153-10-2023
Published online: 15 January 2024
Abstract
Suppose that $K_3$ is a non-normal cubic extension over the rational field $\mathbb Q$. We study the asymptotic distribution of the coefficients $a_{K_3}(n)$ of the Dedekind zeta-function $\zeta _{K_3}(s)$ over the sparse sequence $\{n_1^2+n_2^2\}$. More precisely, we establish an asymptotic formula for the $l$th power of $a_{K_3}(n)$ over the sequence $\{n_1^2+n_2^2\}$ on average, where $n_1, n_2, l \in \mathbb Z$ and $l\geq 2$. In particular, when $2\leq l \leq 8$ we improve the previous results. Moreover, the asymptotic formula for the variance of $a_{K_3}^l(n)$ for $1\leq n\leq x$, $n=a^2+b^2$ is also established.
