I'm very interested in possible randomness in prime numbers distribution. There are many methods for
"decomposition" regularity and randomness in primes (e.g. subtraction of some asymptotics , analysis Riemann Zeta zeroes instead of primes etc. ).
But some trend always remains;
or (if we try to get finite range of data, like here), the data fail randomness tests, e.g. for equidistribution.
Recently I play with Möbius function. Its values {-1,0,1} are not equally distributed;
so let's make a new function: parity of the number of distinct primes in some integer.
If $n=p_1^{a_1} \cdot p_2^{a_2} ... p_{\omega (n)}^{a_{\omega (n)}}$
then
$$\mu_{my} (n) = \begin{cases}
+1, & \text{if }\omega (n)\text{ is even} \\
-1, & \text{if }\omega (n)\text{ is odd}
\end{cases}$$
It seems values of $\mu_{my} (n)$ are really random (obey many randomness tests and have no any trend).
So I'd like to know, are there some theorems or conjectures about randomness in "Möbius-like" functions?
There is a similar variant. If the numbers n with moebius(n) = 0 are omitted, the moebius-function behaves as a random-walk if and only if the riemann-hypothesis is true. As there is a strong evidence that the riemann-hypothesis is true, you can use the moebius-funtion with a high probability as a superb random generator.