Density questions on arithmetic equivalence

TL;DR

This paper improves understanding of when two number fields are arithmetically equivalent by establishing a positive density threshold for prime types, and shows that finitely many zeta function coefficients suffice for verification.

Contribution

It demonstrates that the density zero condition can be strengthened to a positive density depending only on the degree of the fields, and provides bounds for the number of zeta coefficients needed.

Findings

The density threshold c_n = 1/(4n^2) guarantees arithmetic equivalence.

A heuristic suggests c_n could be improved to 2/n^2.

Finitely many zeta function coefficients suffice to determine arithmetic equivalence.

Abstract

It is a classic result that two number fields have equal Dedekind zeta functions if and only if the arithmetic type of a prime $p$ is the same in both fields for almost all prime $p$. Here, almost all means with the possible exception of a set of Dirichlet density zero. One of the results of this paper shows that the condition density zero can be improved to a specific positive density that depends solely in the degree of the fields. More specifically, for every positive $n$ we exhibit a positive constan $c_{n}$ such that any two degree $n$ number fields $K$ and $L$ are arithmetically equivalent if and only if the set of primes $p$ such that the arithmetic type of $p$ in $K$ and $L$ is not the same has Dirichlet density at most $c_n$. We in fact show that $\displaystyle c_n=\frac{1}{4n^2}$ works and give a heuristic evidence that points to the fact that this value might be improved to $\displaystyle \frac{2}{n^2}$. We also show that to check whether or not two number fields are arithmetically equivalent it is enough to check equality between finitely many coefficients of their zeta functions, and we give an upper bound for such number.