5. Special correlations
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Problem 5.1.
Show that \begin{equation*} \left| \sum_{1 \leq n < N} \lambda(n) \lambda(N-n) \right| < N-1. \end{equation*}-
Remark. [Alex de Faveri] This has been proved by Sacha Mangerel (https://arxiv.org/abs/2404.12117) for all $N \geq N_0$, where $N_0$ is an ineffective constant.
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Problem 5.2.
Show that \begin{equation*} \left| \mathbb{E}_{n \in \mathbb{N}}^{\log} \lambda(n^2+1) \right| < 1 - \epsilon, \end{equation*} and that \begin{equation*} \left| \mathbb{E}_{p \in \mathbb{P}} \lambda(p-1) \right| < 1 - \epsilon \end{equation*} for some $\epsilon > 0$. It would even be interesting to show that both sets $\{p : \lambda(p-1) = 1 \}$ and $\{p : \lambda(p-1) = -1 \}$ are infinite. For the first bound, using the entropy decrement argument, wouldn’t it be enough to show that $\left| \sum_{p \approx P} \sum_{n \approx X} \lambda(n^2 +p^2) \right| = o\left(X\frac{P}{\log P}\right)$ with $P = \log \log \log X$? -
Problem 5.3.
What can we say about correlations with itself of \begin{equation*} \mathbb{1}_{p^+(n) \leq n^\delta}, \end{equation*} with $\delta \to 0$? Here $p^+(n) := \max\{p: p|n\}$. Even upper bounds would be welcome. This kind of result would imply sharp mean values for Dirichlet polynomials supported on smooth numbers.
Cite this as: AimPL: Sarnak's conjecture, available at http://aimpl.org/sarnakconjecture.