Loading Web-Font TeX/Main/Regular
| Register
\newcommand{\Cat}{{\rm Cat}} \newcommand{\A}{\mathcal A} \newcommand{\freestar}{ \framebox[7pt]{$\star$} }

5. Special correlations


    1. Add remark

      Problem 5.1.

      Show that \begin{equation*} \left| \sum_{1 \leq n < N} \lambda(n) \lambda(N-n) \right| < N-1. \end{equation*}
        1. 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.
            • Remark. [Sacha Mangerel] It has been shown in the published version of the aforementioned paper (https://academic.oup.com/imrn/advance-article/doi/10.1093/imrn/rnae149/7704606?utm_source=advanceaccess&utm_campaign=imrn&utm_medium=email&login=false) that the above bound holds for all N \geq 11, and only fails when N is 2,3,5 or 10.


                • Add remark

                  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?


                    • Add remark

                      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.