Odd order cases of the logarithmically averaged Chowla conjecture

Tao, Terence; Teräväinen, Joni

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

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.

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.

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

[1] Bourgain, Jean; Sarnak, Peter; Ziegler, Tamar Disjointness of Moebius from horocycle flows, From Fourier analysis and number theory to Radon transforms and geometry (Developments in Mathematics), Volume 28, Springer, 2013, pp. 67-83 | DOI | MR | Zbl

[2] Chowla, Sarvadaman The Riemann hypothesis and Hilbert’s tenth problem, Mathematics and its Applications, 4, Gordon and Breach Science Publishers, 1965 | MR | Zbl

[3] Elliott, Peter D. T. A. On the correlation of multiplicative functions, Notas Soc. Mat. Chile, Volume 11 (1992) no. 1, pp. 1-11 | MR | Zbl

[4] Frantzikinakis, Nikos Ergodicity of the Liouville system implies the Chowla conjecture, Discrete Anal., Volume 2017 (2017), 19, 41 pages (Art. ID 19, 41 pages) | MR | Zbl

[5] Frantzikinakis, Nikos An averaged Chowla and Elliott conjecture along independent polynomials, Int. Math. Res. Not., Volume 2018 (2018) no. 12, pp. 3721-3743 | MR | Zbl

[6] Frantzikinakis, Nikos; Host, Bernard The logarithmic Sarnak conjecture for ergodic weights, Ann. Math., Volume 187 (2018) no. 3, pp. 869-931 | DOI | MR | Zbl

[7] Frantzikinakis, Nikos; Host, Bernard; Kra, Bryna Multiple recurrence and convergence for sequences related to the prime numbers, J. Reine Angew. Math., Volume 611 (2007), pp. 131-144 | MR | Zbl

[8] Friedlander, John; Iwaniec, Henryk Opera de cribro, Colloquium Publications, 57, American Mathematical Society, 2010 | MR | Zbl

[9] Green, Ben; Tao, Terence An inverse theorem for the Gowers $U^3$-norm, with applications, Proc. Edinb. Math. Soc., Volume 51 (2008) no. 1, pp. 73-153 | DOI | Zbl

[10] Green, Ben; Tao, Terence Linear equations in primes, Ann. Math., Volume 171 (2010), pp. 1753-1850 | DOI | MR | Zbl

[11] Green, Ben; Tao, Terence The Möbius function is strongly orthogonal to nilsequences, Ann. Math., Volume 175 (2012) no. 2, pp. 541-566 | DOI | Zbl

[12] Green, Ben; Tao, Terence; Ziegler, Tamar An inverse theorem for the Gowers $U^{s+1}\left[N\right]$-norm, Ann. Math., Volume 176 (2012) no. 2, pp. 1231-1372 | DOI | MR | Zbl

[13] Hardy, Godfrey H.; Littlewood, John E. Some problems of ‘Partitio numerorum’; III: On the expression of a number as a sum of primes, Acta Math., Volume 44 (1923) no. 1, pp. 1-70 | DOI | MR | Zbl

[14] Hoeffding, Wassily Probability inequalities for sums of bounded random variables, J. Am. Stat. Assoc., Volume 58 (1963), pp. 13-30 | DOI | MR | Zbl

[15] Kátai, Imre A remark on a theorem of H. Daboussi, Acta Math. Hung., Volume 47 (1986), pp. 223-225 | DOI | MR | Zbl

[16] Le, Anh Ngoc Nilsequences and multiple correlations along subsequences (2017) (https://arxiv.org/abs/1708.01361) | Zbl

[17] Leibman, Alexander Nilsequences, null-sequences, and multiple correlation sequences, Ergodic Theory Dyn. Syst., Volume 35 (2015) no. 1, pp. 176-191 (corrected version available at people.math.osu.edu/leibman.1/preprints/msqx.pdf) | DOI | MR | Zbl

[18] Matomäki, Kaisa; Radziwiłł, Maksym Multiplicative functions in short intervals, Ann. Math., Volume 183 (2016) no. 3, pp. 1015-1056 | DOI | MR | Zbl

[19] Matomäki, Kaisa; Radziwiłł, Maksym; Tao, Terence An averaged form of Chowla’s conjecture, Algebra & Number Theory, Volume 9 (2015), pp. 2167-2196 | DOI | MR | Zbl

[20] Tao, Terence The logarithmically averaged Chowla and Elliott conjectures for two-point correlations, Forum Math. Pi, Volume 4 (2016), 8, 36 pages (Art. ID e8, 36 pages) | MR | Zbl

[21] Tao, Terence Equivalence of the logarithmically averaged Chowla and Sarnak conjectures, Number theory—Diophantine problems, uniform distribution and applications, Springer, 2017, pp. 391-421 | Zbl

[22] Tao, Terence; Teräväinen, Joni The structure of logarithmically averaged correlations of multiplicative functions, with applications to the Chowla and Elliott conjectures (2017) (to appear in Duke Math. J., https://arxiv.org/abs/1708.02610) | Zbl

[23] Tao, Terence; Vu, Van H. Additive combinatorics, Cambridge Studies in Advanced Mathematics, 105, Cambridge University Press, 2006 | MR | Zbl