The Liouville function is the completely multiplicative function with for every prime . The Chowla conjecture predicts that this function behaves randomly. Here is a version of this conjecture.

This conjecture received a lot of attention in the last decade, since Peter Sarnak derived from it an orthogonality criterion for the Liouville (or, similarly, the Mobius function), now known as the “Sarnak conjecture”. Both Chowla’s and Sarnak’s conjectures remain open, however many weaker versions and special cases have been derived (see, for instance, this survey). Perhaps the most impressive result so far is the following theorem of Tao, which was in turn the starting point for many of the more recent developments.

Tao’s main theorem in his paper is actually significantly more general than Theorem 2, for it allows more general affine maps inside the argument of , and in fact replaces with a much wider class of multiplicative functions. To see why Theorem 2 follows from Conjecture 1, observe that any sequence satisfying

must also satisfy

In the terminology and notation of my earlier post, the conclusion of this theorem states that the uniform logarithmic average of the sequence is , i.e.,

for .

In this post I will present Tao’s proof restricted to the following special case. (Most of the proof remains unchanged in the general case, but the presentation becomes easier when we avoid general ).

We will use a correspondence principle to reformulate the problem in the language of dynamical systems to, among other things, simplify the notation. This approach was also carried out by Tao in his blog, and the main motivation for me to write this post was to try to use an easier construction of a Furstenberg system, which would still keep track of the “congruence classes of the points” in the system, as detailed below the fold.