On a Tur\'an's theorem for arithmetic progressions

Abstract

Let $m\geq 1$ be a fixed integer, $a$ an integer satisfying $(a,m)=1$, and $z\geq 1$ a real parameter. Denote by $\omega_{z}(n;m,a)$ the number of distinct prime divisors $p$ of $n$ satisfying $p\equiv a\, (m)$ and $p\leq z$. We study an asymptotic behaviour of $\sum_{n\leq x}\left(\omega_{z}(n;m,a)-\frac{1}{\varphi(m)}\log\log z\right)^{k}$ as $x\to\infty$ for a wide range of positive integer $k\geq 2$, where $\varphi(\cdot)$ is the Euler function. Following a method of Granville and Soundararajan we lead an asymptotic formula for the above. Also, we investigate $\sum_{n\leq x}\left(\omega(n;m,a)-\frac{1}{\varphi(m)}\log\log x\right)^{k}$, where $\omega(n;m,a)$ denotes the number of distinct prime divisors $p$ of $n$ such that $p\equiv a\, (m)$.