Instytut Matematyczny Polskiej Akademii Nauk / Institute of Mathematics / Publishing house / Journals and Serials / Acta Arithmetica / Online First articles

Abstract

Let $K_1$ and $K_2$ be two number fields of degree $k$, and let $a_{K_i}(p)$ denote the ideal counting function in the Dedekind zeta function of $K_i$. The fields $K_1$ and $K_2$ are called arithmetically equivalent if they share the same Dedekind zeta function. We investigate the analytic density of the set $S$ of primes $p$ for which $a_{K_1}(p) \neq a_{K_2}(p)$ when $K_1$ and $K_2$ are not arithmetically equivalent. We confirm a conjecture of Mantilla-Soler (2023) by showing that this analytic density is at least $2/k^2$. We also study density problems associated with $a_K(p)$ for a single number field $K$.