Let denote the Liouville function. A well-known conjecture of Chowla asserts that, for any distinct natural numbers , one has
as . This conjecture remains unproven for any with . Using the recent results of Matomäki and Radziwiłł on mean values of multiplicative functions in short intervals, combined with an argument of Kátai and Bourgain, Sarnak, and Ziegler, we establish an averaged version of this conjecture, namely
as , whenever goes to infinity as and is fixed. Related to this, we give the exponential sum estimate
as uniformly for all , with as before. Our arguments in fact give quantitative bounds on the decay rate (roughly on the order of ) and extend to more general bounded multiplicative functions than the Liouville function, yielding an averaged form of a (corrected) conjecture of Elliott.
"An averaged form of Chowla's conjecture." Algebra Number Theory 9 (9) 2167 - 2196, 2015. https://doi.org/10.2140/ant.2015.9.2167