The concept is founded on:
• The fact that proving the occurrence of the Möbius function's values of 1 and -1 to be completely random would affirm the truth of the Riemann Hypothesis.
• The premise that the randomness of a selected set of numbers can be determined using the so-called density of Euler’s Phi function of that set.
Further details about the idea of the density of Euler's Phi function and its relationship with the entropy of number sets are discussed in my two previous posts (link here and at the beginning of this post). However, to condense it into a single sentence: the density of Euler's Phi function for a given set of numbers (i.e., the sum of the Phi function values of all elements of the set divided by the sum of the set) behaves in a very specific and predictable manner, depending on how methodically or randomly the elements of the set were selected. The density of the entire set of natural numbers equals 1/zeta(2), so 6/pi².
If we choose any subset of natural numbers in a completely random manner, its density will be identical. The more methodically we make the selection, the more the density of that set will deviate from this value (either increasing or decreasing).
It is noteworthy, although not central to this discussion, that a number set reaches both maximum and minimum levels of entropy at the same density value of the Phi function—a completely random set has a density identical to that of the entire set of natural numbers, which is not random at all.
Assuming full rigor that this property (determining the randomness of a set through the density of the Phi function) is true, it can be used to assess the randomness in the occurrence of individual values of the Möbius function.
Demonstrating that the occurrence of the values 1 and -1 of the Möbius function is completely random would confirm the truth of the RH, which in our case means (based on the above assumption) demonstrating that the Phi densities for both Möbius function values (1 and -1, considered separately) are identical and thereby equal their joint density (calculated together for 1 and -1).
It turns out that, "as expected," the behavior of the Phi density for specific Möbius function values is very regular, and for the values 0, and -1 and 1 together, it's easy to calculate.
The Phi function density for nonzero values naturally equals the probability of the occurrence of so-called carefree couples, which is equal to
The obvious question is whether this density is the same for the values 1 and -1 separately. Observation (calculations within the range possible to be conducted by an ordinary computer) confirms that this is the case, and they show that these densities converge quickly to this value even for relatively small, arbitrarily chosen intervals of numbers.
Of course, observation alone is not proof, but this can be demonstrated by using the idea I wrote some time ago based on analyzing the behavior and variability of what I call the 'relative' Möbius function (here is a link to the document describing this idea, which could itself be a basis for attempting to prove the Riemann Hypothesis, but now, there are significantly more arguments for this).
Here comes a very important "definition" - relative because its values depend only on prime numbers starting from 2 to a specific p. The behavior of this function in the range from 0 to p!^2 (the square of the primorial for a given p) is very specific, and it allows for concrete calculations, including the frequency and distribution of occurrences of individual values of this function. As p approaches infinity, the specified values obviously converge with those for the "regular" Möbius function.
What I am doing now (which is not covered by the previous document) is using the analysis of the behavior of the relative Phi function density, that is, analyzing how the Phi function density behaves for specific values of the relative Möbius function. Thus, to calculate the values of the relative Phi function, we take (similarly as in the case of the relative Möbius function) only its prime divisors from the range 0 to a given p.
Because the pattern of occurrence of individual values of the relative Möbius function is symmetric around the center of the range 0 to p!^2, this gives us the possibility of precisely and relatively easily calculating the density of the relative Phi function for these values. These formulas are algorithmically simple but very extensive (the number of elements depends on the number of combinations of ways to choose n-element sets from the set of all primes up to a given p), so I do not include them here (I present here the outline, the idea of the proof, not the proof itself).
Calculating the value of the relative Phi function for a given p using these formulas, we observe what we might call an obvious effect: that for small p, the density of the relative Phi function approaches the expected value (previously mentioned 0.4282…) for values of the relative Möbius function equal to -1 more quickly than for 1 (initially, odd-numbered terms predominate, as described in the attached document regarding the frequency of these values for the relative Möbius function), but as p increases, the densities for both values of the relative Möbius function (1 and -1) converge, as expected, to the same value equal to the aforementioned infinite product = 0.4282...
The next key point - It can be demonstrated (again, not presenting the entirety of the argument here due to its breadth but relative simplicity) that the previously mentioned elaborate formulas (for a chosen, finite p) simplify as p approaches infinity for both values -1 and 1 of the relative Möbius function to the earlier mentioned formula. This, in turn, proves that the Phi density (now "normal") for the "normal" values of the Möbius function is indeed identical for both values (-1 and 1, each individually) and equals this infinite product = 0.4282…
Accepting the initial assumption implies that the occurrence of values 1 and -1 in the Möbius function is entirely random, which in turn substantiates the truth of the Riemann Hypothesis.
Summarizing the entire idea in key points:
1. I assert that the density of the Phi function of subsets of natural numbers (and other specific sets of numbers) can be an indicator of their randomness – this directly stems from the way the Phi density of subsets chosen in a specific (non-random) manner behaves. This is partially demonstrated by my earlier argument about the formulas for the Phi density of sets in the form n+/-p.
2. We have a formula for the density of the Phi of the nonzero values of the Möbius function equal to an infinite product:
3. We have formulas for the frequency of occurrence of zero and nonzero values (both 1 and -1 separately) of the relative Möbius function from zero to the square of the primorial of a given p, and correspondingly the densities of the relative Phi function for these sets.
4. These formulas, as p approaches infinity, can be “simplified” into a form where both (both for values 1 and -1) converge to the same value, which equals the Phi density for the nonzero values (1 and -1 together) of the “normal” Möbius function.
5. Thus, we can say that we have derived formulas for the Phi densities for values 1 and -1 (separately for each), which are equal to each other and to their combined density, which in turn means
6. We can assert that the probabilities of occurrence of values 1 and -1 of the Möbius function are equal, and their occurrence is entirely random.
7. This, in turn, implies the truth of the Riemann Hypothesis.
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