On a Tur\'an's theorem for small primes

Abstract

Denote by $\omega(n)$ the number of distinct prime divisors of the natural number $n$. In 2007, Granville and Soundararajan gave a quite new method to compute the higher moments $\sum_{n\leq x}(\omega(n)-\log\log x)^{k}$, for a wide range of integers $k\geq 2$. In this notes, we shall apply the method for $\omega_{z}(n)$ which denotes the number of distinct prime divisors $\leq z$ of $n$. Especially, for odd integers $k\geq 3$, we lead asymptotic formulas for $\sum_{n\leq x}(\omega_{z}(n)-\log\log z)^{k}$, under certain restrictions on $k$, $z$, and $x$. Also, we refer to a framework of the approach.