Siegel Zeros and Sarnak's Conjecture

Abstract

Assuming the existence of Siegel zeros, we prove that there exists an increasing sequence of positive integers for which Chowla's Conjecture on $k$-point correlations of the Liouville function holds. This extends work of Germ\'an and K\'atai, where they studied the case $k=2$ under identical hypotheses. An immediate corollary, which follows from a well-known argument due to Sarnak, is that Sarnak's Conjecture on M\"obius disjointness holds. More precisely, assuming the existence of Siegel zeros, there exists a subsequence of the natural numbers for which the Liouville function is asymptotically orthogonal to any sequence of topological entropy zero.