Here is a very nice paper http://www.ams.org/journals/tran/2010-362-12/S0002-9947-2010-05235-3/S0002-9947-2010-05235-3.pdf which led me to thinking about the problems below.
Define the Liouville function for $A \subseteq \mathbb{P}$ by $\lambda_{A}(n) = (-1)^{\Omega_A(n)}$, where $\Omega_A(n)$ is the number of prime factors of $n$ coming from $A$ counting multiplicity. Now denote $L_A(x) := \sum_{n \leq x} \lambda_A(n)$.
Now let's define another function $F$ by $F(n) = \min \{L_A(n) : A \subseteq \mathbb{P}\}$. First few values of $F$ are $1, 0, -1, 0, -1, 0, -1, -2, -1, 0, -1, -2, -3, -4, -3, -2, -3, -4, -5, -4, -5, -4, -5, -6, -5, -6, -7, -8, -9$ And here is a graph for small values:
My algorithm for calculating $F$ is as follows:
Consider all possible values on prime numbers not bigger than $\sqrt{n}$. For every other prime we know whether we want to set $-1$ or $1$ on it, because it depends only on sum of values on numbers smaller than $\sqrt{n}$.
So the complexity is $O\left(n 2^{\frac{2\sqrt{n}}{\log{n}}}\right)$.
Here are my questions regarding this function:
- is $F$ always non-positive for $n > 1$? [Edit: Yes, it is]
- does it reach $0$ infinitely many times? [Edit: No, it doesn't]
- what is $F$'s asymptotic? Is it better than $O(\frac{n}{\log n})$?
- how to calculate $F$ efficiently?
- what is relation between $F$ and Riemann Hypothesis?
I am also interested in function $L_{\mathbb{P} \setminus \{2\}}$. All questions above can be applied also to this function, but answers to the fourth and fifth are the same as for the standard $L$.
I believe this function should be much easier to investigate, because we can use similar methods as in investigating Polya's conjecture.
Here is a graph of $L$ (green), $L_{\mathbb{P} \setminus \{3\}}$ (light blue), $L_{\mathbb{P} \setminus \{2\}}$ (red) and $L_{\mathbb{P} \setminus \{2, 3\}}$ (dark blue):
