Sarnak's conjecture

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6. Unbounded multiplicative functions

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Problem 6.1.
Let $(X, \mu, T)$ be a totally uniquely ergodic $0$ -entropy system. We know from recent work of Frantzikinakis-Host that if $g:\N \to \C$ is a bounded multiplicative function, then for all $f \in C(X)$ with $\int_X f d \mu = 0$ and $x \in X$ we have

$\begin{equation*} \mathbb{E}^{\log}_{n \in \mathbb{N}} g(n) f(T^nx) = 0. \end{equation*}$

Now, if $g: \N \to \C$ is a (possibly unbounded) multiplicative function, show that for all $f \in C(X)$ , $x \in X$ and fixed $c \in \C$ ,

$\begin{equation*} \mathbb{E}^{\log}_{n \in \mathbb{N}} \mathbb{1}_{g(n)=c} f(T^nx) = \mathbb{E}^{\log}_{n \in \mathbb{N}} \mathbb{1}_{g(n)=c} \cdot \int_X f d \mu. \end{equation*}$ This can be done for totally ergodic nilsystems.
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Problem 6.2.
Can anything be done for correlations of unbounded functions? What is the behavior in almost all intervals of $\begin{equation*} \sum_{x \leq n \leq x + H} g(n), \end{equation*}$ with $g$ an unbounded multiplicative function such that $g(p) = O(1)$ , and $H = \log^A x$ for $A>0$ fixed?
1. Remark. [Sacha Mangerel] This problem is addressed in the following preprint: https://arxiv.org/abs/2108.11401, assuming $|g(p)|$ is not too sparsely supported, and $g$ is bounded by a suitable (generalized) divisor function. The method is an adaptation of the work of Matomaki-Radziwill in https://arxiv.org/abs/2007.04290. The exponent $A$ in the range $H = (\log x)^A$ depends on the growth of $\sum_{p \leq X} (|g(p)|-1)^2p^{-1}$ .

Cite this as: AimPL: Sarnak's conjecture, available at http://aimpl.org/sarnakconjecture.