A generalization of an Erdős inequality connected to n!

Summary.

In 1946, P. Erdős remarked the following inequality connected to n!. Write $n! = \prod_{p}p^{w_{p}(n)}$. Whatever the primes p and q, if w _p (n) > w _q (n), then $p^{w_{p}(n)} > q^{w_{q}(n)}$. The notion of factorials has been generalized to every number field and, more generally, by M. Bhargava, to every subset of a Dedekind domain. We give here a characterization of the Dedekind domains such that Erdős inequalities hold and we apply it in particular to quadratic and cyclotomic fields. We also prove that Bhargava’s factorials associated to the set of prime numbers satisfy Erdős inequalities.