is equivalent to the prime number theorem, while the improved (and essentially best possible) rate of convergence
for every positive is equivalent to the Riemann hypothesis.
Of particular interest are statements involving correlations (or lack thereof) between and certain “structured” functions. In this direction, Green and Tao established “orthogonality” of with any nilsequence. More precisely, they showed that for any nilpotent Lie group , any discrete subgroup for which is compact, any Lipschitz function , any and any ,
which was an important ingredient in their proof of the “finite complexity” version of the Dickson/Hardy-Littlewood prime tuple conjecture. Perhaps inspired by this result, Sarnak then conjectured that in fact the Liouville function (actually, the closely related Möbius function) is orthogonal to every deterministic sequence:
Conjecture 1 (Sarnak conjecture for the Liouville function) Let be a compact metric space and let be a homeomorphism. If the topological dynamical system has topological entropy, then for every continuous and every ,
Sarnak’s conjecture is still open, despite some remarkable progress (see the recent survey of Ferenczi, Kułaga-Przymus and Lemańczyk for information on the progress made so far).
A useful tool to establish partial cases of Sarnak’s conjecture (and other results not implied by the Sarnak conjecture) is the following orthogonality criterion of Katai.
Remark 1 One can in fact relax the condition on to be only a multiplicative function. This allows the criterion to be applied, for instance, with being the Möbius function.
The statement of Theorem 2 is reminiscent of the van der Corput trick, which is a ubiquitous tool in the fields of uniform distribution and ergodic Ramsey theory, was the protagonist of a survey Vitaly Bergelson and I wrote and has been mentioned repeatedly in this blog. In fact one can prove both results (Katai’s orthogonality criterion and van der Corput’s trick) using the same simple functional analytic principle. In this post I present this principle and use it to motivate proofs for the two theorems.