# Sparser variance for primes in arithmetic progression

## TL;DR

This paper derives an asymptotic formula for the variance of prime counts in arithmetic progressions with moduli restricted to specific sequences, extending Montgomery-Hooley results to new classes of moduli.

## Contribution

It introduces a novel asymptotic formula for prime variance in progressions with moduli defined by functions like $t^c$ and exponential forms, expanding understanding of prime distribution.

## Key findings

- Asymptotic formula for prime variance in new moduli sequences
- Extension of Montgomery-Hooley formula to non-integer power and exponential moduli
- Improved understanding of prime distribution in specialized arithmetic progressions

## 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 $F(t) = t^c$ ($c > 1$, $c \not\in \mathbb{N}$) or $F(t) = \exp\big((\log t)^{\gamma}\big)$ ($1 < \gamma < 3/2$).