2016/05/16 by Terence Tao, Tao, Terence · 3 citations
Mathematics · #11N37 #Advanced Mathematical Identities #Analytic Number Theory Research #Benford’s Law and Fraud Detection #FOS: Mathematics #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.1605.04628
openalex publication_date 2016/05/16 · openalex created_date 2022/10/03 · openalex updated_date 2026/07/28
Let λ denote the Liouville function. The Chowla conjecture asserts that ∑n ≤ X λ(a1 n + b1) λ(a2 n+b2) … λ(ak n + bk) = oX → ∞(X) for any fixed natural numbers a1,a2,…,ak and non-negative integer b1,b2,…,bk with aibj-ajbi ≠ 0 for all 1 ≤ i < j ≤ k, and any X ≥ 1. This conjecture is open for k ≥ 2. As is well known, this conjecture implies the conjecture of Sarnak that ∑n ≤ X λ(n) f(n) = oX → ∞(X) whenever f : \bf N → \bf C is a fixed deterministic sequence and X ≥ 1. In this paper, we consider the weaker logarithmically averaged versions of these conjectures, namely that ∑X/ω≤ n ≤ X (λ(a1 n + b1) λ(a2 n+b2) … λ(ak n + bk))/(n) = oω→ ∞(log ω) and ∑X/ω≤ n ≤ X (λ(n) f(n))/(n) = oω→ ∞(log ω) under the same hypotheses on a1,…,ak,b1,…,bk and f, and for any 2 ≤ ω≤ X. Our main result is that these latter two conjectures are logically equivalent to each other, as well as to the "local Gowers uniformity" of the Liouville function. The main tools used here are the entropy decrement argument of the author used recently to establish the k=2 case of the logarithmically averaged Chowla conjecture, as well as the inverse conjecture for the Gowers norms, obtained by Green, Ziegler, and the author.