THE LOGARITHMICALLY AVERAGED CHOWLA AND ELLIOTT CONJECTURES FOR TWO-POINT CORRELATIONS

By Terence Tao. Forum of Mathematics, Pi, 2015

125 citations recorded by the source corpus.

Let $\unicode[STIX]{x1D706}$ denote the Liouville function. The Chowla conjecture, in the two-point correlation case, asserts that $$\begin{eqnarray}\mathop{\sum }_{n\leqslant x}\unicode[STIX]{x1D706}(a_{1}n+b_{1})\unicode[STIX]{x1D706}(a_{2}n+b_{2})=o(x)\end{eqnarray}$$ as $x\rightarrow \infty$ , for any fixed natural numbers $a_{1},a_{2}$ and nonnegative integer $b_{1},b_{2}$ with $a_{1}b_{2}-a_{2}b_{1}\neq 0$ . In this paper we establish the logarithmically averaged version $$\begin{eqnarray}\mathop{\sum }_{x/\unicode[STIX]{x1D714}(x)<n\leqslant x}\frac{\unicode[STIX]{x1D706}(a_{1}n+b_{1})\unicode[STIX]{x1D706}(a_{2}n+b_{2})}{n}=o(\log \unicode[STIX]{x1D714}(x))\end{eqnarray}$$ of the Chowla conjecture as $x\rightarrow \infty$ , where $1\leqslant \unicode[STIX]{x1D714}(x)\leqslant x$ is an arbitrary function of $x$ that goes to infinity as $x\rightarrow \infty$ , 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 $\unicode[STIX]{x1D706}$ , 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.

Subfields: Algebra and Number Theory, Discrete Mathematics and Combinatorics

View Paper: https://www.semanticscholar.org/paper/fd5bdb2f9e98ffb2efa17cb88315abfb03c87742

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Citations in the corpus

Almost all orbits of the Collatz map attain almost bounded values (2019, background): “For technical reasons, the notion of ‘almost all’ that we will use here is based on logarithmic density, which has better approximate multiplicative invariance properties than the more familiar notion of natural density (see [20] for a related phenomenon in a more number-theoretic context).”
Sarnak’s Conjecture: What’s New (2017, background): “The starting point of the paper [68] is a surprising Tao’s identity (implicit in [154]) for general sequences which in its ergodic theory language (cf.”
The structure of logarithmically averaged correlations of multiplicative functions, with applications to the Chowla and Elliott conjectures (2017, methodology): “A different kind of averaging of the k = 1 case of the Elliott conjecture was considered by the first author in [36].”
The logarithmic Sarnak conjecture for ergodic weights (2017, methodology): “We give a sketch of the proof of a more general identity in Appendix C; the argument is almost entirely based on the argument given by Tao in [65].”