Odd order cases of the logarithmically averaged Chowla conjecture

Tao, Terence; Teräväinen, Joni

Journal de théorie des nombres de Bordeaux, Volume 30 (2018) no. 3, pp. 997-1015.

A famous conjecture of Chowla states that the Liouville function $\lambda \left(n\right)$ has negligible correlations with its shifts. Recently, the authors established a weak form of the logarithmically averaged Elliott conjecture on correlations of multiplicative functions, which in turn implied all the odd order cases of the logarithmically averaged Chowla conjecture. In this note, we give a new proof of the odd order cases of the logarithmically averaged Chowla conjecture. In particular, this proof avoids all mention of ergodic theory, which had an important role in the previous proof.

Une conjecture bien connue de Chowla affirme que les corrélations des translatés de la fonction $\lambda \left(n\right)$ de Liouville sont asymptotiquement nulles. Dans un article récent, les auteurs ont démontré un résultat partiel en direction de la conjecture d’Elliott logarithmiquement pondérée concernant les corrélations des fonctions multiplicatives, qui à son tour implique tous les cas de la conjecture de Chowla avec un nombre impair de translatés. Dans cet article, nous donnons une nouvelle démonstration de ce dernier résultat sur la conjecture de Chowla. En particulier, celle-ci évite l’usage de la théorie ergodique qui joue un rôle crucial dans notre démonstration précédente.

Classification: 11N37
Keywords: Liouville function, Chowla’s conjecture, Gowers uniformity norms

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