Sparser variance for primes in arithmetic progression

Abstract

We obtain an analog of the Montgomery-Hooley asymptotic formula for the variance of the number of primes in arithmetic progressions. In the present paper the moduli are restricted to the sequences of integer parts $[F(n)]$, where $F(t)=t^c$ ($c>1$, $c\notin \mathbb{N}$) or $F(t)=\exp ((\log t{)}^{\gamma })$ ($1<\gamma <3/2$).