On a Turán's theorem for arithmetic progressions

Abstract

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

arXiv comment38 pages