A Note on the least squarefree number in an arithmetic progression
Ramon M. Nunes
TL;DR
This paper establishes an improved asymptotic formula for counting squarefree numbers in arithmetic progressions with squarefree moduli, advancing previous results by Prachar through novel estimates for solutions of congruences.
Contribution
It introduces a new estimate for counting solutions of congruences that surpasses the Weil bound, enabling more precise asymptotic formulas for squarefree numbers in progressions.
Findings
Derived an asymptotic formula for squarefree numbers in arithmetic progressions.
Developed a new estimate for solutions of congruences inside a box.
Enhanced previous bounds by Prachar for squarefree number distribution.
Abstract
We prove an asymptotic formula for squarefree in arithmetic progressions with squarefree moduli, improving previous results by Prachar. The main tool is an estimate for counting solutions of a congruence inside a box that goes beyond what can be obtained by using the Weil bound.
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