Higher order uniformity of the Möbius function
Also appears in collection : Additive Combinatorics / Combinatoire additive
The Liouville function $\lambda(n)$ takes the value +1 or -1 depending on whether $n$ has an even or an odd number of prime factors. The Liouville function is closely related to the characteristic function of the primes and is believed to behave more-or-less randomly. I will discuss my very recent work with Radziwill, 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 number of sign patterns for the Liouville sequence and to a new averaged version of Chowla’s conjecture.
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