Sign changes of the Liouville function in arithmetic progressions

TL;DR

This paper proves that within certain bounds, the Liouville function takes both positive and negative values in the same arithmetic progression, extending understanding of sign changes in number theory.

Contribution

It establishes the existence of integers with opposite Liouville function signs in the same residue class within explicit bounds related to the modulus.

Findings

Existence of integers m, n ≤ q^{5/2 + ε} with m ≡ n ≡ a mod q

Liouville function takes both signs in the same residue class

Results relate to explicit bounds similar to Linnik's theorem

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 $p \ll q^{5.5}$.