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Rencontres de théorie analytique des nombres

Explicit constructions of multiplicative functions with small correlations

Salle Grisvard, IHP, Paris

There is a well-known relationship between the distribution of primes and distribution of $\pm 1$ signs of the Liouville function (the completely multiplicative function taking the value $-1$ at all primes). A conjecture of Chowla, analogising the Hardy-Littlewood prime $k$-tuples conjecture, predicts that the autocorrelations of $\lambda$, e.g. $\frac{1}{x} \sum_{n \leq x} \lambda_{n+1} \cdots\lambda_{n+k}$ tend to 0 on average as $x$ tends to $\infty$. This conjecture, along with its generalisation to the broader collection of bounded "non-pretentious" multiplicative functions, due originally to Elliott, remain wide open for $k \geq 2$. Previously, there were no explicit examples in the literature of (deterministic and scale-independent) non-pretentious multiplicative functions known to satisfy Elliott's conjecture. In this talk I will present a construction of a non-pretentious multiplicative function $f : \mathbf{N} \rightarrow \{-1,1\}$ all of whose auto-correlations tend to 0 on average, answering a(n ergodic theory) question of Lemanczyk and de la Rue. I will further discuss the following applications of this construction:

i) a proof that Chowla's conjecture does not imply the Riemann Hypothesis, i.e., there are $\pm 1$-valued multiplicative functions $f$ all of whose autocorrelations tend to 0, but that do not exhibit square-root cancellation on average (the object of some recent speculation);

ii) there are multiplicative subsemigroups of $\mathbf{N}$ with Poissonian gap statistics, thus giving an unconditional multiplicative analogue of a classical result of Gallagher about primes in short intervals.

(Joint work with Oleksiy Klurman, Par Kurlberg and Joni Teravainen.)