2023/04/11 by Klurman, Oleksiy, Mangerel, Alexander P., Teräväinen, Joni
#11N37 #37A34 #Dynamical Systems (math.DS) #FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.2304.05344
Let f:ℕ→ \mathbbD be a multiplicative function. Under the merely necessary assumption that f is non-pretentious (in the sense of Granville and Soundararajan), we show that for any pair of distinct integer shifts h1,h2 the two-point correlation (1)/(x)∑n≤ xf(n+h1)f(n+h2) tends to 0 along a set of x∈ℕ of full upper logarithmic density. We also show that the same result holds for the k-point correlations (1)/(x)∑n≤ xf(n+h1)⋯ f(n+hk) if k is odd and f is a real-valued non-pretentious function. Previously, the vanishing of correlations was known only under stronger non-pretentiousness hypotheses on f by the works of Tao, and Tao and the third author. We derive several applications, including: (i) A classification of ± 1-valued completely multiplicative functions that omit a length four sign pattern, solving a 1974 conjecture of R.H. Hudson. (ii) A proof that a class of "Liouville-like" functions satisfies the unweighted Elliott conjecture of all orders, solving a problem of de la Rue. (iii) Constructing examples of multiplicative f:ℕ→ \-1,0,1\ with a given (unique) Furstenberg system, answering a question of Lemańczyk. (iv) A density version of the Erdős discrepancy theorem of Tao.