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?

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.

• Could you please give some references about the above? I know only about "Prime numbers and the Riemann hypothesis" by Mazur and Stein, but this is mostly a popular science book (and a bit on the verbose side, too). I am looking for a rigorous (or at leasts mostly rigorous) text. Commented Jan 23, 2023 at 12:23

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.