Linnik's problem for the Möbius function

We show that the least natural number having an odd number of prime
factors and belonging to any arithmetic progression $a \pmod q$ is
bounded by $q^{2+o(1)}$. This can be seen as a multiplicative analogue of Linnik's
problem on the least prime in an arithmetic progression. This is based on
joint work with Kaisa Matomäki.