Speaker: Kaisa Matomaki
26 August, 2020 13:30 UTC
Abstract: The Liouville function takes a value +1 or -1 at a natural number $n$ depending on whether $n$ has an even or an odd number of prime factors. The Liouville function is believed to behave more or less randomly. In particular a famous conjecture of Sarnak says that the Liouville function does not correlate with any sequence of “low complexity” whereas a longstanding conjecture of Chowla says that the Liouville function has negligible correlations with its own shifts. I will discuss conjectures of Sarnak and Chowla and my very recent work with Radziwiłł, Tao, Teräväinen, and Ziegler, where we show that, in almost all intervals of length $X^\varepsilon$, the Liouville function does not correlate with polynomial phases or more generally with nilsequences. I will also discuss applications to superpolynomial word complexity for the Liouville sequence and to a new averaged version of Chowla’s conjecture.