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We investigate the distribution of the function $\omega(n)$, the number ofdistinct prime divisors of $n$, in residue classes modulo $q$ for naturalnumbers $q$ greater than 2. In particular we ask `prime number races' stylequestions, as suggested by Coons and Dahmen in their paper `On the residueclass distribution of the number of prime divisors of an integer'.

Residue races of the number of prime divisors function