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There are several recent results on non-trivial bounded functions $f$ that are orthogonal, i.e. $$\sum_{n\leq x} \mu(n)f(n)=o(x),$$ to the Mobius function $\mu(n)$. These include the results of Green and Tao on Nilsequences (which includes information on the history of the problem) and Green on bounded depth circuits. The lecture notes by Peter Sarnak on Mobius randomness is an interesting overview of the subject.

Sarnak mentions there that it is easy to construct counter examples, or examples which decay very slowly. Of course, its easy to do so if $f$ is sparsely supported or has partial sums that are the inverse Mellin transform of a function having positive powers of $\zeta(s)$ as factors.

What are some examples of bounded, non-vanishing (or at least positive density) functions that are not orthogonal to $\mu(n)$, and which don't come from $\zeta(s)$ in an obvious way?

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    $\begingroup$ This is a tautological answer, but: one could take any function that is the product of $\mu(n)$ and an arbitrary function that has nonzero mean on the squarefree integers. $\endgroup$ – Terry Tao Nov 14 '16 at 17:11
  • $\begingroup$ Do any explicit examples come from positive entropy flows? I believe P. Sarnak said we know such functions exist, but perhaps he meant that $\mu$ is used in their construction. $\endgroup$ – Kevin Smith Nov 14 '16 at 21:10
  • $\begingroup$ Sure; take the left shift on the orbit closure of the Mobius sequence $\mu$, viewed as a point in $\{-1,0,+1\}^n$. $\endgroup$ – Terry Tao Nov 14 '16 at 21:20
  • $\begingroup$ Perhaps this was a tautological question! Thanks though. $\endgroup$ – Kevin Smith Nov 14 '16 at 22:07
  • $\begingroup$ @Terry Tao: is $(-1)^n$ known to have a non zero mean on the squarefrees? $\endgroup$ – Kevin Smith Nov 27 '16 at 15:33
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This question has been considered by Lemanczyk and others, and Lemanczyk developed a quite general way to produce dynamical systems which are not disjoint from Mobius (unfortunately or fortunately, depending on ones preferences, they are of positive entropy). The basic idea is to use the square free flow constructed by Sarnak (which Terry have mentioned) and to find some symbolic system which "approximates" this flow for large times. See for example the detailed survey - https://arxiv.org/abs/1410.1673 In the end, the fact that there are systems which are not Mobius disjoint is not that surprising - Sarnak has computed the entropy of the Mobius flow and show it positive, general ergodic theory (i.e. the construction of the Bernoulli factor by Sinai/Ornstein) would tell you it is "easy" to correlate with Mobius.

Just another related interesting comment - Bourgain has constructed a system with positive entropy which is Mobius disjoint.

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  • $\begingroup$ Thank you Asaf. Is it clear yet what distinguishes those positive entropy systems which are and are not Mobius disjoint? $\endgroup$ – Kevin Smith Nov 18 '16 at 15:30
  • $\begingroup$ Great Question! I don't know of anything specific and I am not intimately familiar with Bourgain's example (just heard it from Peter a while ago). One problem is that the Bernoulli factor is highly "measure theortical" construction and you will not encounter that in topological systems (hence this is possible to be Mobius disjoint with positive entropy), but considering Lemanczyk's general approach, I suspect you will be able to produce examples for very general sequences, without appealing into arithmetics, but there should be some natural threshold, maybe half of the maximal entropy. $\endgroup$ – Asaf Nov 20 '16 at 8:04

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