Mertens's formula, k-full numbers and sum of two squares

Abstract

In this article we prove generalizations of the well-known Mertens's formula. We also define some arithmetic functions related to the exponents of the primes in the prime factorization of the positive integers and study its average. Finally, we study square-free numbers, s-full numbers and perfect powers not exceeding x which can be represented as sums of two squares.