The structure of logarithmically averaged correlations of multiplicative functions, with applications to the Chowla and Elliott conjectures

Abstract

Let g 0 , . . . , g k : N → D be 1-bounded multiplicative functions, and let h 0 , . . . , h k ∈ Z be shifts. We consider correlation sequences f : N → Z of the formwhere 1 ≤ ω m ≤ x m are numbers going to infinity as m → ∞, and lim is a generalised limit functional extending the usual limit functional. We show a structural theorem for these sequences, namely that these sequences f are the uniform limit of periodic sequences f i . Furthermore, if the multiplicative function g 0 . . . g k "weakly pretends" to be a Dirichlet character χ, the periodic functions f i can be chosen to be χ-isotypic in the sense that f i (ab) = f i (a)χ(b) whenever b is coprime to the periods of f i and χ, while if g 0 . . . g k does not weakly pretend to be any Dirichlet character, then f must vanish identically. As a consequence, we obtain several new cases of the logarithmically averaged Elliott conjecture, including the logarithmically averaged Chowla conjecture for odd order correlations. We give a number of applications of these special cases, including the conjectured logarithmic density of all sign patterns of the Liouville function of length up to three, and of the Möbius function of length up to four.

Correlations of multiplicative functions along deterministic and independent sequences

“…Our first theorem extends results of Tao [23] and Tao, Teräväinen [26] that correspond to the case a(n) = n (see Section 2 for definitions of the notions used). Theorem 1.1.…”

“…Using the results of the previous subsections it is easy to deduce results on sign patterns attained by multiplicative functions. The next result extends [26, Corollary 1.10(i)], which corresponds to the case where a(n) = n (the result in [26] is stated only for f = λ but the argument given works in the more general setup of the next theorem). Theorem 1.6.…”

“…The conjecture is settled for ℓ = 1, this is elementarily equivalent to the prime number theorem, and remains open for ℓ ≥ 2. Recently, a version involving logarithmic averages was established for ℓ = 2 by Tao [23] and for all odd values of ℓ by Tao and Teräväinen [25,26]. Similar results are also known for Cesàro averages on almost all scales [27].…”

“…This structural result is used implicitly in the disjointness statement of Theorem 3.3 and gives the identities of Theorem 3.1. These identities allow us to deduce Theorem 1.1 from the main results in [23] and [26].…”

“…• If we restrict to sequences that take values on the unit circle, then ∼ is an equivalence relation and a ∼ b is equivalent to E log p∈P |a(p) − b(p)| 2 = 0. • Using terminology from [26] we have that two multiplicative functions f, g : N → U satisfy f ∼ g exactly when "f weakly pretends to be g".…”