Author

Haruka Sakai

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Preprint Sep 2026

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

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)...

T. Minamide, Haruka Sakai, Y. Tanigawa · 0 citations

Preprint Sep 2026

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

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...

T. Minamide, Haruka Sakai, Y. Tanigawa · 1 citation