Fourier uniformity of bounded multiplicative functions in short intervals on average

Abstract

<p>Let λ denote the Liouville function. We show that as X→∞ ,</p><p>∫2XXsupα∣∣∣∣∑x</p><p>for all H≥Xθ with θ>0 fixed but arbitrarily small. Previously, this was only known for θ>5/8 . For smaller values of θ this is the first “non-trivial” case of local Fourier uniformity on average at this scale. We also obtain the analogous statement for (non-pretentious) 1-bounded multiplicative functions. We illustrate the strength of the result by obtaining cancellations in the sum of λ(n)Λ(n+h)Λ(n+2h) over the ranges h</p>