Randomness in prime numbers

Question

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?

Answer 1

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.

Answer 2

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.