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## 3. Möbius and Liouville systems

1. #### Problem 3.1.

Can a Liouville system be isomorphic to $(\mathbb{T}^2, m_{\mathbb{T}^2}, T)$, where $T:\mathbb{T}^2\to \mathbb{T}^2$ is given by $\left(\begin{smallmatrix} x\\ y \end{smallmatrix}\right) \mapsto \left(\begin{smallmatrix} x+y\\ y \end{smallmatrix}\right)$ mod $1$?
• #### Problem 3.2.

Is $-1$ in the spectrum of a Liouville system? Equivalently, can we show that \begin{equation*} \sum_{n\leq X} (-1)^{n-1} \frac{\lambda(n+h_1) \dots \lambda(n+h_k)}{n} = o(\log X)? \end{equation*}
1. Remark. [Nikos] More generally, one can ask to show that $e(p/q)$ is not in the spectrum of a Liouville system when $p/q$ is not an integer. If this is shown, one immediate consequence (using Szemerédi’s theorem) is that on the range of the Liouville function there are arbitrarily large patterns $++\ldots+$ and $--\ldots-$ (see Problem 2.2).
• #### Problem 3.3.

Let $X_{\mu^2} = \overline{\{T^k(\mu^2) : k \in \N\}}$ and $X_{\mu} = \overline{\{T^k(\mu) : k \in \N\}}$, for $T$ the shift map. What are the pre-images of $f:X_{\mu} \to X_{\mu^2}$ given by $(x_i) \mapsto (x_i^2)$ in $X_{\mu^2}$?

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