# Interesting examples of functions that are not orthogonal to the Mobius function?

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?

• 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. – Terry Tao Nov 14 '16 at 17:11
• 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. – Kevin Smith Nov 14 '16 at 21:10
• Sure; take the left shift on the orbit closure of the Mobius sequence $\mu$, viewed as a point in $\{-1,0,+1\}^n$. – Terry Tao Nov 14 '16 at 21:20
• Perhaps this was a tautological question! Thanks though. – Kevin Smith Nov 14 '16 at 22:07
• @Terry Tao: is $(-1)^n$ known to have a non zero mean on the squarefrees? – Kevin Smith Nov 27 '16 at 15:33