On two conjectures concerning squarefree numbers in arithmetic progressions

Abstract

We prove upper bounds for the error term of the distribution of squarefree numbers up to X in arithmetic progressions modulo q making progress towards two well-known conjectures concerning this distribution and improving upon earlier results by Hooley. We make use of recent estimates for short exponential sums by Bourgain-Garaev and for exponential sums twisted by the M\"obius function by Bourgain and Fouvry-Kowalski-Michel.

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