Sign changes of the Liouville function in arithmetic progressions

We show that for any $\epsilon >0$, prime $q$ sufficiently large with respect to $1\mathrm{/}\epsilon$ and residue class $(a,q)=1$, there exist two integers $m,n\le q^{5\mathrm{/}2+\epsilon }$ with $m\equiv n\equiv a(\mathrm{m}\mathrm{o}\mathrm{d}q)$ such that $\lambda (m)=-1$ and $\lambda (n)=+1$, where $\lambda$ denotes the Liouville function. Our result is motivated by Heath-Brown’s explicit exponent in Linnik’s theorem, establishing the existence of primes $p\equiv a(\mathrm{m}\mathrm{o}\mathrm{d}q)$ with $p\ll q^{5.5}$.