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1. Lower bounds


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      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.