Squarefree integers in large arithmetic progressions

TL;DR

This paper improves the understanding of how squarefree integers are distributed in large arithmetic progressions with prime moduli by establishing a higher exponent of distribution using advanced exponential sum bounds.

Contribution

It introduces a new upper bound for bilinear exponential sums, surpassing traditional methods, to achieve a better exponent of distribution for squarefree numbers.

Findings

Exponent of distribution ≥ 2/3 + 1/57 for squarefree numbers in prime moduli

New bilinear sum bounds surpass Polya-Vinogradov method

Improved distribution results from 1958 Prachar's work

Abstract

We show that the exponent of distribution of the sequence of squarefree numbers in arithmetic progressions of prime modulus is $\geq 2/3 + 1/57$, improving a result of Prachar from 1958. Our main tool is an upper bound for certain bilinear sums of exponential sums which resemble Kloosterman sums, going beyond what can be obtained by the Polya-Vinogradov completion method.