Distribution of squarefree integers with double congruence conditions

Abstract: Given a positive integer $M$ with reduced residue system $Φ_M$, we partition the set of squarefree integers according to some congruence conditions on the number of prime factors in each class $φ\in Φ_M$, and prove two relevant equidsitribution results in this setting. We discover the bias phenomenon in some special cases, and formulate a conjecture with numerical support.