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1. Lower bounds


    1. Add remark

      Problem 1.1.

      Let $h \in \N$ be fixed. Is there a sequence $X_k \to \infty$ such that \begin{equation*} \left| \sum_{n \leq X_k} \lambda(n) \lambda(n + h) \right| \to \infty \end{equation*} as $k \to \infty$? We could also consider variants for multiplicative functions $f$ with values in $\{\pm 1\}$. We can handle $f$ pretentious, and can also do it assuming $4$-point Chowla or the existence of arbitrarily long strings of $+1$’s for $\lambda$. Would a Sudoku-type argument work here? If $h = h_k$ is allowed to vary with $k$, then we can probably show this as well.


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          Problem 1.2.

          [P. Sarnak] Is there a bounded sequence $a(n)$ such that each $a(n)$ is computable in time $O(\log^A n)$ for some fixed $A > 0$ and such that

          \begin{equation*} \left| \sum_{n \leq X} \lambda(n) a(n) \right| > \epsilon X \end{equation*} for all $X$ and some fixed $\epsilon > 0$? Can we say something about the Furstenberg systems of such an $a(n)$? In particular, do they have positive entropy?


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              Problem 1.3.

              Can we obtain lower bounds for

              \begin{equation*} \int_X^{2X} \left| \sum_{x\leq n\leq x+H} \lambda(n) \right| d x? \end{equation*} In particular, can we get a lower bound of the form $XH^{\frac{1}{2}-\epsilon}$?


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                  Problem 1.4.

                  Can we get a reasonable omega result for \begin{equation*} \left| \sum_{n \leq X} f(n) \right|, \end{equation*} where $f: \N \to \{\pm1\}$ is totally multiplicative? Anything better than $\log \log \log \log \log X$ would be interesting.
                    1. Remark. [J. Teräväinen] This problem was solved by R. McNamara in his PhD thesis [https://escholarship.org/uc/item/4wr015m0], giving a growth rate of $(\log \log X)^{1/484-o(1)}$. H. Helfgott and M. Radziwiłł outline in [https://arxiv.org/abs/2103.06853] a significant improvement to the exponent of $\log \log X$.

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