An average theorem for tuples of k-free numbers in arithmetic progressions

TL;DR

This paper derives an asymptotic formula for the variance of tuples of k-free numbers in arithmetic progressions, extending classical results to a new setting involving multiple number tuples.

Contribution

It introduces a novel asymptotic formula for the variance of k-free number tuples in arithmetic progressions, inspired by Montgomery-Hooley and Barban-Davenport-Halberstam theorems.

Findings

Established an asymptotic formula for the variance of k-free tuples in progressions

Extended classical variance results to tuples of numbers

Provided new insights into the distribution of k-free numbers in arithmetic progressions

Abstract

We obtain an asymptotic formula, in the spirit of the Montgomery-Hooley refinement of the Barban-Davenport-Halberstam Theorem, for the variance associated with tuples of k-free numbers in arithmetic progressions.