  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.
