On the distribution of the coefficients of the Dedekind zeta function over a certain sequence of positive integers

Let $𝕂$ be a nonnormal algebraic number field of cubic degree given by the polynomial $x^{3} + a x^{2} + b x + c$ of discriminant $D_{𝕂} < 0$ . We study the asymptotic formulae for the sums

\sum_{\begin{array}{c} n = \sum_{i = 1}^{6} n_{i}^{2} \leq x \\ (n_{1}, n_{2}, n_{3}, n_{4}, n_{5}, n_{6}) \in ℤ^{6} \end{array}} a_{𝕂}^{2} (n) and \sum_{\begin{array}{c} n = \sum_{i = 1}^{6} n_{i}^{2} \leq x \\ (n_{1}, n_{2}, n_{3}, n_{4}, n_{5}, n_{6}) \in ℤ^{6} \end{array}} a_{𝕂}^{3} (n),

where $a_{𝕂} (n)$ is number of integral ideals of norm $n$ and we improve the earlier known error terms.

Keywords

Dedekind zeta function, Hecke eigenforms, normalized Fourier coefficient, symmetric square $L$-function, Dirichlet character

Mathematical Subject Classification

Primary: 11F11, 11F30, 11M06, 11R42

Milestones

Received: 17 January 2026

Revised: 19 March 2026

Accepted: 21 April 2026

Published: 30 May 2026

Authors

Matrapu S. Datt
School of Mathematics and Statistics University of Hyderabad Hyderabad India

Ekta Soni
School of Mathematics and Statistics University of Hyderabad Hyderabad India

Ayyadurai Sankaranarayanan
School of Mathematics and Statistics University of Hyderabad Hyderabad India

Vol. 15, No. 3, 2026

Matrapu S. Datt, Ekta Soni and Ayyadurai Sankaranarayanan

Abstract

Keywords

Mathematical Subject Classification

Milestones

Authors