Sign changes of the Liouville function in arithmetic progressions

Abstract

We show that for any $\varepsilon > 0$, prime $q$ sufficiently large with respect to $1 / \varepsilon$ and residue class $(a,q) = 1$, there exist two integers $m, n \leq q^{5/2 + \varepsilon}$ with $m \equiv n \equiv a \pmod{q}$ 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 $p \equiv a \pmod{q}$ with $p \ll q^{5.5}$.