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6. Unbounded multiplicative functions


    1. Add remark

      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.


        • Add remark

          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.