Ergodicity of the Liouville system implies the Chowla conjecture

Ergodicity of the Liouville system implies the Chowla conjecture, Discrete Analysis 2017:19, 41 pp.

The Liouville function $\lambda:\mathbb N\to\{-1,1\}$ takes a product $p_1p_2\dots p_k$ of (not necessarily distinct) primes $p_1,\dots,p_k$ to $(-1)^k$. That is, $\lambda(n)$ is the parity of the number of primes in the prime decomposition of $n$. It is a completely multiplicative function (that is, $\lambda(mn)=\lambda(m)\lambda(n)$ for any two positive integers $m,n$, and like various other multiplicative functions it plays an important role in number theory. In particular, it appears to behave like a random $\pm 1$ sequence in various ways, and if this could be proved rigorously it would have major consequences: the Riemann hypothesis, for instance, is equivalent to the statement that the sums $\lambda(1)+\lambda(2)+\dots+\lambda(n)$ grow more slowly than $n^\alpha$ whenever $\alpha>1/2$. (Note that this would be the expected behaviour for a random sequence.) It is known at least that $\lambda(1)+\dots+\lambda(n)=o(n)$, and in fact this statement is equivalent to the prime number theorem.

Of great interest recently has been the study of more detailed statistical properties of the Liouville function, where one is interested not just in individual values but in correlations between nearby values. For example, it is believed that $\lambda(x)$ and $\lambda(x+1)$ should be independent, in the sense that $\lambda(1)\lambda(2)+\dots+\lambda(n)\lambda(n+1)=o(n)$. More generally, the Chowla conjecture asserts that for any set $\{k_1,\dots,k_r\}$ of distinct positive integers we have that $\sum_{m\leq n}\lambda(m+k_1)\dots\lambda(m+k_r)=o(n)$. This conjecture is still wide open, though weakenings of it, which have been proved in the last few years by Matomäki, Radziwiłł, Tao and Teräväinen (in various combinations), turn out to be very useful for applications. For example, Tao, building on work by Matomäki and Radziwiłł, established a logarithmically averaged version of the conjecture for two-point correlations, and used it in his solution of the Erdõs discrepancy problem. Also, Tao and Teräväinen have proved a logarithmically averaged version for all correlations of an odd number of points, which, slightly surprisingly, turns out to be an easier problem. (One might at first think that this would imply the same for all correlations. However, the statement that $\lambda(m+k_1)\dots\lambda(m+k_r)$ averages zero is not the same as the statement that the values $\lambda(m+k_1),\dots,\lambda(m+k_r)$ are independent, and it does not imply independence even if one assumes the same statement for all subsets of $\{k_1,\dots,k_r\}$ of odd size, or indeed for all subsets of $\mathbb N$ of odd size.)

It turns out to be rather fruitful to reformulate questions of this type as questions about dynamical systems. A standard construction in symbolic dynamics is to take an infinite sequence of letters in a finite alphabet and take the closure of all its shifts (in the product topology). The shift acts on this closure, turning it into a dynamical system. Thanks to the Furstenberg correspondence principle, it comes with a natural measure that makes it into a measure-preserving system. The properties of this system relate to various features of the sequence, and methods from ergodic theory and topological dynamics can be used to prove interesting theorems that do not involve dynamics in their statements.

The main result of this paper is that the general logarithmically averaged Chowla conjecture (that is, the statement for all correlations and not just correlations between pairs) follows from a dynamical statement that looks significantly weaker. In dynamical terms, the Chowla conjecture states that the dynamical system one obtains from the Liouville sequence has the Bernoulli property, which means that it is isomorphic to the system one gets from all $\pm 1$ sequences. (This implication holds because if every finite string occurs with the right frequency, then shifts of the Liouville sequence land in a basic open neighbourhood with the right frequency, so all the basic open neighbourhoods have the right measure and the system becomes the same as what one would get with the shift acting on the space of all $\pm 1$ sequences with the product measure.) The paper shows that to prove the Bernoulli property, it is enough to establish that the system is ergodic, which means that there are no invariant subsets of measure strictly between 0 and 1. In general, an implication like this is far from true: there are a number of properties that say that a dynamical system is “somewhat random”: the property of being ergodic is one of the weakest, and the Bernoulli property is the strongest. So the result is telling us something interesting about the Liouville sequence. Several powerful tools from analytic number theory, additive combinatorics, and ergodic theory are used in the proof.