Distinct exponents in the prime factorization

Abstract

Following Erd\H{o}s (1982) and Sanna (2019), we study the arithmetic function $h(n)$, which is defined to be the number of distinct exponents in the prime factorization of a positive integer $n$. Among other things, we show that $$ \sum_{n\leq x}h(\phi(n)) \asymp x\left(\frac{\log\log x}{\log\log\log x}\right)^{1/2}, $$ where $\phi$ is the Euler totient function. The key ingredient is the Poisson random model for $\omega(n,T)$, the number of the prime divisors of $n$ in a given subset of primes $T$, which was introduced in a recent work of Ford.