Extended Riemann Connections (4)

First just a slight correction on a formula already given!

To recap, I suggested that, for example ,in calculating the relative frequency of numbers where two or more prime factors occurs at most 2 times, that we multiply the initial result - where one or more factors occur at most 2 times (i.e. 1/ζ(3) – 1/ζ(2) = .224 approx.) - by a common ratio that is half of the original result i.e. {1/ζ(3) – 1/ζ(2)}/2 = .112 approx..

So therefore the calculated result for the frequency of two or more prime factors occurring at most 2 times = .224 * .112 = .025 approx.

Therefore, again, we would expect about 5 in every 200 numbers to belong to this particular prime factor category! And from extensive numerical investigation, this tallies very accurately with empirical results.

However, I then suggested that one would keep employing multiplication by this common ratio, in calculating further results (where primes occur at most 2 times).

So to calculate the frequency of numbers where 3 or more prime factor occur at most 2 times, this would imply a result of .025 * .112 ... = .0028 (i.e. slightly less than 3 per 1000).

However repeated empirical investigation kept suggesting half of this result i.e. .0014 approx.

So therefore it now appears that the ratio should be further divided by 2 for each subsequent calculation.

Therefore to calculate the frequency of numbers, where 4 or more primes occur at most 2 times (in the factor composition of the number), we would therefore multiply the previous result by {1/ζ(3) – 1/ζ(2)}/8.

So the - very small - calculated result would be .0014 * .0028 = .00000392 though proper empirical confirmation of such results becomes very difficult at this low level of factor frequency!  

Therefore in general terms the ratio to calculate where (n + 1) factors occur exactly 2 times we multiply the result where n or more factors occur at most 2 times by {1/ζ(3) – 1/ζ(2)}/2^{n – 1}.


Now, we have already seen that where 1 or more prime factors can occur at most 3 times that the initial result = ζ(4) – 1/ζ(3) = .092 (approx).

Then to calculate where 2 or more prime factors can occur at most 3 times, we multiply the initial result by the ratio {1/ζ(4) – 1/ζ(3)}/4 = .023.

Therefore the result = .092 * .023 = .002116. This worked out at slightly more than 2 per 1000 which appears fully consistent with empirical results.

However to then move on to calculate - the very low - relative frequency where 3 or more prime factors occur at most 3 times, we multiply the previous result by{1/ζ(4) – 1/ζ(3)}/8.

The result therefore = .002116  * .0115  = .000024 (approx).

And just one more case to illustrate!

The relative frequency where one or more prime factors occurs at most 4 times = 1/ζ(5) – 1/ζ(4) = .04.

Then to calculate the relative frequency where 2 or more prime factors can occur at most 4 times, we multiply the previous result by {1/ζ(5) – 1/ζ(4)}/8 = .005 (approx).

Therefore, we obtain .04 * .005 = .0002 (approx) i.e. about 2 in 10,000.

However for the - considerably less frequent - case where 3 or more prime factors occur at most 4 times we now multiply the previous result by {1/ζ(5) – 1/ζ(4)}/16 = .0002 * .0025 = .0000005 (approx).

So for the more general case where prime factors can occur at most n times, we start by dividing the initial result 1/ζ(n + 1) – 1/ζ(n) - relating to the relative frequency where 1 or more factors occur at most n times - by {1/ζ(n + 1) – 1/ζ(n)}/2^{n – 1} to obtain the initial adjusting ratio.

Therefore the relative frequency where 2 or more factors can occur at most n times,

= 1/ζ(n + 1) – 1/ζ(n) * {1/ζ(n + 1) – 1/ζ(n)}/2^{n – 1}.

Then for subsequent calculations we keep further dividing the adjusting ratio in each case by 2!


We are now ready to move on to the next stage of our investigations.

Earlier, I drew attention to a remarkable feature of the number system!

It would appear - and indeed there is a deep theoretical reason why this should be so - that what I have referred to in the past as the Zeta 2 function likewise applies here in its most basic expression.

Now again the very rationale of my dynamic approach is that two complementary functions work in parallel in generating the features of the number system.

The first one is the well-known Riemann zeta function (which I refer to as the Zeta 1 function).

ζ₁(s₁) = 1/1^s1 + 1/2^s1 + 1/3^s1 + 1/4^s1 +…

This could be equally written as

ζ₁(s₁) = 1^{– s1}  + 2^{– s1}  + 3^{– s1}  + 4^{– s1}  +…

So on our recent investigations, we have been showing how this function (defined for the positive integer values of s₁), is so hugely relevant in explaining the manner in which the frequency of the various prime factor combinations of numbers is distributed throughout the number system.

However I have long maintained that that an equally important complementary function is intimately associated with the Riemann function.

This is what I refer to as the Zeta 2 function and is defined as

ζ₂(s₂) = 1 + s₂¹ + s₂² + s₂³ + s₂⁴ +….

Notice the inverse (complementary) nature of both functions!

In the Zeta 1, the natural numbers are defined in base terms i.e. as the number (that is raised to an exponent or power); in the Zeta 2 the natural numbers are defined in dimensional terms (to which the base numbers are raised).

Likewise the unknown values of s (s₁ and s₂ respectively) represent dimensional powers in Zeta 1, whereas they represent base numbers in Zeta 2.

So again, Zeta 1 has been used to explain the relative frequency with which - in general external terms - different combinations of prime factors occur.

However when it then comes to the internal calculation of the combined number of factors associated with the different prime combinations, a specific version of the Zeta 2 is now required.

Once more we have seen that 1/2 of all factors belong to the category (Class 1) where numbers are composed of prime factors where each occurs just once.

Put another way this equally entails that 1/2 of all factors belong to the other general category where 1 or more primes can occur more than once in the factor composition of numbers.

And this in itself is a simple remarkable fact regarding the number system representing - ultimately - an exact balance as between prime independence (where no factors occur more than once) and prime interdependence (where 1 or more primes occur more than once)!

Indeed in more philosophical terms this represents an exact balance as between key quantitative (analytic) and qualitative (holistic) features of the number system.

And notice that this balance relates directly to 1/2, which equally plays a central role with respect to the Riemann Hypothesis. In fact properly understood, from a philosophic mathematical perspective (that properly recognises the inherent dynamic nature of the number system), this points to the fact that the Riemann Hypothesis states the key condition required to ensure the mutual consistency as between both quantitative and qualitative aspects of the number system.  

Then we were able to demonstrate more than this! For we were able to further show that 1/4 of all factors belong to the Class 2 category (where 1 or more primes can occur at most 2 times).

Continuing on, 1/8 of all factors then belong to the Class 3 category (where 1 or more primes can occur at most 3 times).

And in general terms 1/2ⁿ of all factors belong to the Class n category (where 1 or more primes can occur at most n times).

So what determines the frequency of the count of factors belonging to each class are progressive terms in the simple geometric series 1, 1/2, 1/4, 1/8, 1/16,… (that conform to the Zeta 2 function where s₂ = 1/2)!

And we will see a much more detailed use of the complementary Zeta 2 function in future blog entries.