Randomness in prime numbers 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?
 A: 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.
A: Distribution of primes completely determined by the following statement:
Positive integers which do not appear in both arrays $A1(i,j)=6i^2+(6i−1)(j−1)$ and $A2(i,j)=6i^2+(6i+1)(j−1)$:
                    |  6   11    16     21   ...|
        A1(i,j) =   | 24   35     46    57   ...|
                    | 54   71     88   105   ...|
                    | 96  119    142   165   ...|
                    |...  ...  ...   ...     ...|


                     |  6    13   20    27   ...|
         A2(i,j) =   | 24    37   50    63   ...|
                     | 54    73   92   111   ...|
                     | 96   121  146   171   ...|
                     |...   ...  ...   ...   ...|

are indexes $k$ of primes in the sequence $S1(k)=6k−1$.
Positive integers which do not appear in both arrays $A3(i,j)=6i^2−2i+(6i−1)(j−1)$ and $A4(i,j)=6i^2+2i+(6i+1)(j−1)$:
                           | 4       9     14       19.. |
                           |20      31     42       53...|
                           |48      65     82       99...|
                  A3(i,j)= |88     111     134     157...|
                           |...   ...      ...     ...   |

                    | 8      15      22     29 ..|
                    |28     41       54     67...|
           A4(i,j)= |60     79       98     117..|
                    |104   129      154    179...|
                    |...    ...     ...     ...  | 

are indexes $k$ of primes in the sequence $S2(k)=6k+1$. Since all primes (except 2 and 3) are in one of two forms $6k−1$ or $6k+1$, so we can find primes simply by picking up positive integers which do not appear in these arrays.(C++ code see http://www.planet-source-code.com/vb/scripts/BrowseCategoryOrSearchResults.asp?lngWId=3&blnAuthorSearch=TRUE&lngAuthorId=21687209&strAuthorName=Boris%20Sklyar&txtMaxNumberOfEntriesPerPage=25
From the above statement it's  obvious that in distribution of primes there is no any kind of randomness.
