Effizient berechenbare Funktion als Gegenbeispiel zu Sarnaks Mobius-Vermutung

Kürzlich haben Gil Kalai und Dick Lipton einen schönen Artikel über eine interessante Vermutung geschrieben, die von Peter Sarnak, einem Experten für Zahlentheorie und Riemannsche Hypothese, vorgeschlagen wurde.

Vermutung. Sei die Möbius-Funktion . Angenommen, ist eine Funktion mit Eingabe in Form einer binären Darstellung von$\mu(k)$$f: \mathbb{N} \to \{-1,1\}$$\mathsf{AC}^0$$k$$k$, then

$$\sum_{k \leq n} \mu(k) \cdot f(k) = o(n) \text.$$

Note that if $f(k) = 1$ then we have an equivalent form of the Prime number theorem.

UPDATE: Ben Green on MathOverflow provide a short paper which claims to prove the conjecture. Take a look at the paper.

On the other hand, we know that by setting $f(k) = \mu(k)$ (with slightly modification so the range is in $\\{-1,1\\}$), the resulting sum has the estimation

What is the lowest complexity class $\mathsf{C}$ we currently know, such that a function $f(k)$ in $\mathsf{C}$ satisfies the estimation

$$\sum_{k \leq n} \mu(k) \cdot f(k) = \Omega(n) \text{?}$$ In particular, since some of the theorists believed that computing $\mu(k)$ is not in $\mathsf{P}$, can we provide other $\mathsf{P}$ functions $f(k)$ which implies a linear growth in the summation? Can even better bounds be obtained?

There have been interesting developments on this problem, however Replacing $AC^0$ with ACC(2) (Namely allowing mod 2 gates as well) is still well out-of-reach. Some progress beyond Ben Green's theorem can be found in this MO question https://mathoverflow.net/questions/57543/walsh-fourier-transform-of-the-mobius-function as well as this one https://mathoverflow.net/questions/97261/mobius-randomness-of-the-rudin-shapiro-sequence . In addition, Jean Bourgain proved Mobius randomness for every monotone function $f$ (in terms of the binary-digit expansion).