An averaged form of Chowla's conjecture

By Kaisa Matomaki, Maksym Radziwill, Terence Tao. 2015

162 citations recorded by the source corpus.

Let $\lambda$ denote the Liouville function. A well known conjecture of Chowla asserts that for any distinct natural numbers $h_1,\dots,h_k$, one has $\sum_{1 \leq n \leq X} \lambda(n+h_1) \dotsm \lambda(n+h_k) = o(X)$ as $X \to \infty$. This conjecture remains unproven for any $h_1,\dots,h_k$ with $k \geq 2$. In this paper, using the recent results of the first two authors on mean values of multiplicative functions in short intervals, combined with an argument of Katai and Bourgain-Sarnak-Ziegler, we establish an averaged version of this conjecture, namely $$\sum_{h_1,\dots,h_k \leq H} \left|\sum_{1 \leq n \leq X} \lambda(n+h_1) \dotsm \lambda(n+h_k)\right| = o(H^kX)$$ as $X \to \infty$ whenever $H = H(X) \leq X$ goes to infinity as $X \to \infty$, and $k$ is fixed. Related to this, we give the exponential sum estimate $$ \int_0^X \left|\sum_{x \leq n \leq x+H} \lambda(n) e(\alpha n)\right| dx = o( HX )$$ as $X \to \infty$ uniformly for all $\alpha \in \mathbb{R}$, with $H$ as before. Our arguments in fact give quantitative bounds on the decay rate (roughly on the order of $\frac{\log\log H}{\log H}$), and extend to more general bounded multiplicative functions than the Liouville function, yielding an averaged form of a (corrected) conjecture of Elliott.

Subfields: Algebra and Number Theory, Discrete Mathematics and Combinatorics

View Paper: https://www.semanticscholar.org/paper/99e96ae72e3b32df5b166ba12ba64dcc46fb231d

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Algebra and Number Theory

In Discrete Mathematics and Combinatorics

Citations in the corpus

THE LOGARITHMICALLY AVERAGED CHOWLA AND ELLIOTT CONJECTURES FOR TWO-POINT CORRELATIONS (2015, methodology): “, bk parameters; see [6, 8, 23] for some recent results in this direction.”
Sarnak’s Conjecture: What’s New (2017, methodology): “In [154], Tao proves the logarithmic version of Chowla conjecture for the correlations of order 2 (which we formulate for the Liouville function): Theorem 3.15 ([154]).”
Automorphisms with quasi-discrete spectrum, multiplicative functions and average orthogonality along short intervals (2015, result): “(n)ν(n) −→ 0 when K → ∞. It has been noted to us by N. Frantzikinakis that the particular case P(n) = αn, which we present in Section 4.1, follows from a recent result of Matoma¨ki, Radziwlli and Tao [27]. We can also reformulate Theorem 4 in the follow ing way. Theorem 5. Assume that ν: N→ C, |ν| ≤”
The structure of logarithmically averaged correlations of multiplicative functions, with applications to the Chowla and Elliott conjectures (2017, background): “The result in [31] was generalised to averages over independent polynomials in several variables in [11].”