Anubhav Sharma ; Ayyadurai Sankaranarayanan - On a general divisor problem related to a certain Dedekind zeta-function over a specific sequence of positive integers

hrj:13033 - Hardy-Ramanujan Journal, February 11, 2024, Volume 46 - 2023 - https://doi.org/10.46298/hrj.2024.13033
On a general divisor problem related to a certain Dedekind zeta-function over a specific sequence of positive integersArticle

Authors: Anubhav Sharma 1; Ayyadurai Sankaranarayanan 2

We investigate the average behavior of coefficients of the Dirichlet series of positive integral power of the Dedekind zeta-function $\zeta_{\mathbb{K}_3}(s)$ of a non-normal cubic extension $\mathbb{K}_3$ of $\mathbb{Q}$ over a certain sequence of positive integers. More precisely, we prove an asymptotic formula with an error term for the sum\[ \sum_{{a_{1}^{2}+a_{2}^{2}+a_{3}^{2}+a_{4}^{2}+a_{5}^{2}+a_{6}^{2}\leq {x}}\atop{(a_{1},a_{2},a_{3},a_{4},a_{5},a_{6})\in\mathbb{Z}^{6}}}a_{k,\mathbb{K}_3} (a_{1}^{2}+a_{2}^{2}+a_{3}^{2}+a_{4}^{2}+a_{5}^{2}+a_{6}^{2}),\]where $(\zeta_{\mathbb{K}_3}(s))^{k}:=\sum_{n=1}^{\infty}\frac{a_{k,\mathbb{K}_3}(n)}{n^{s}}$.


Volume: Volume 46 - 2023
Published on: February 11, 2024
Accepted on: February 11, 2024
Submitted on: February 9, 2024
Keywords: Dedekind zeta-function,Hecke eigenvalues,non-normal cubic extension,[MATH]Mathematics [math]

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