A famous conjecture of Chowla states that the Liouville function 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 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.
Accepted:
Published online:
DOI: 10.5802/jtnb.1062
Keywords: Liouville function, Chowla’s conjecture, Gowers uniformity norms
@article{JTNB_2018__30_3_997_0, author = {Terence Tao and Joni Ter\"av\"ainen}, title = {Odd order cases of the logarithmically averaged {Chowla} conjecture}, journal = {Journal de th\'eorie des nombres de Bordeaux}, pages = {997--1015}, publisher = {Soci\'et\'e Arithm\'etique de Bordeaux}, volume = {30}, number = {3}, year = {2018}, doi = {10.5802/jtnb.1062}, zbl = {1441.11255}, mrnumber = {3938639}, language = {en}, url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1062/} }
TY - JOUR AU - Terence Tao AU - Joni Teräväinen TI - Odd order cases of the logarithmically averaged Chowla conjecture JO - Journal de théorie des nombres de Bordeaux PY - 2018 SP - 997 EP - 1015 VL - 30 IS - 3 PB - Société Arithmétique de Bordeaux UR - https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1062/ DO - 10.5802/jtnb.1062 LA - en ID - JTNB_2018__30_3_997_0 ER -
%0 Journal Article %A Terence Tao %A Joni Teräväinen %T Odd order cases of the logarithmically averaged Chowla conjecture %J Journal de théorie des nombres de Bordeaux %D 2018 %P 997-1015 %V 30 %N 3 %I Société Arithmétique de Bordeaux %U https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1062/ %R 10.5802/jtnb.1062 %G en %F JTNB_2018__30_3_997_0
Terence Tao; Joni Teräväinen. Odd order cases of the logarithmically averaged Chowla conjecture. Journal de théorie des nombres de Bordeaux, Volume 30 (2018) no. 3, pp. 997-1015. doi : 10.5802/jtnb.1062. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1062/
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