The Logarithmically Averaged Chowla and Elliott Conjectures for Two-Point Correlations

Abstract

Let λ denote the Liouville function. The Chowla conjecture, in the two-point correlation case, asserts thatas x → ∞, for any fixed natural numbers a 1 , a 2 and nonnegative integer b 1 , b 2 with a 1 b 2 −a 2 b 1 = 0. In this paper we establish the logarithmically averaged versionof the Chowla conjecture as x → ∞, where 1 ω(x) x is an arbitrary function of x that goes to infinity as x → ∞, thus breaking the 'parity barrier' for this problem. Our main tools are the multiplicativity of the Liouville function at small primes, a recent result of Matomäki, Radziwiłł, and the author on the averages of modulated multiplicative functions in short intervals, concentration of measure inequalities, the Hardy-Littlewood circle method combined with a restriction theorem for the primes, and a novel 'entropy decrement argument'. Most of these ingredients are also available (in principle, at least) for the higher order correlations, with the main missing ingredient being the need to control short sums of multiplicative functions modulated by local nilsequences. Our arguments also extend to more general bounded multiplicative functions than the Liouville function λ, leading to a logarithmically averaged version of the Elliott conjecture in the two-point case. In a subsequent paper we will use this version of the Elliott conjecture to affirmatively settle the Erdős discrepancy problem.

Sign Patterns of the Liouville and Möbius Functions

“…. One can bound this by the probability that γ (k) = γ (k), where γ , γ are selected as in the proof of (15). But if we fix the variables p 1 , .…”

“…The further study of this random graph (or similar such graphs) may have some further applications; in particular, one may hope to use expansion properties of this graph to make progress towards the k = 2 case of the Chowla conjecture (Conjecture 1.3). In fact by pursuing this direction further, the third author has recently shown that a logarithmic form of Chowla's conjecture holds, and consequently that the logarithmic density of integers n for which (µ(n), µ(n + 1)) = (ε 1 , ε 2 ) exists (see [15]). …”

“…This can be bounded by the expectation of w γ , where γ is chosen as in the proof of (15). But from Lemma 7.1, this expectation is at most exp(O(k))/(log log X ) k , which from the choice (12) of k is (barely) O(log −100 X ), which is acceptable.…”

The Erdős discrepancy problem

“…To do this, we use a recent result of the author [21] regarding correlations of multiplicative functions: Theorem 1.10 (Logarithmically averaged nonasymptotic Elliott conjecture). [21, Theorem 1.3] Let a 1 , a 2 be natural numbers, and let b 1 , b 2 be integers such that a 1 b 2 − a 2 b 1 = 0.…”

Harmonic Analysis on the Positive Rationals. Determination of the Group Generated by the Ratios

“…Attention is drawn to application of a (recent) result of Tao [24], that a sufficiency of large sums…”