On a Turán'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 ${\sum }_{n\le x}(\omega (n)-\log \log x{)}^k$, for a wide range of integers $k\ge 2$. In this notes, we shall apply the method for ${\omega }_z(n)$ which denotes the number of distinct prime divisors $\le z$ of $n$. Especially, for odd integers $k\ge 3$, we lead asymptotic formulas for ${\sum }_{n\le x}({\omega }_z(n)-\log \log z{)}^k$, under certain restrictions on $k$, $z$, and $x$. Also, we refer to a framework of the approach.

arXiv comment51 pages