A new characterization of arithmetic equivalence (Q1898684)

Two algebraic number fields \(K\) and \(L\) are said to be equivalent if their Dedekind zeta-functions coincide. It has been known that this is equivalent to the following statement: for every rational prime \(p\) the sets of degrees of prime ideals lying over \(p\) in \(K\) and \(L\) coincide [see the second author, J. Number Theory 10, 489--509 (1978; Zbl 0393.12009)]. The authors prove that if for almost all primes \(p\) (i.e. up to a set of zero Dirichlet density) the number of prime ideals over \(p\) in \(K\) and \(L\) coincide, then these fields are arithmetically equivalent.