Riemann Zeta Function: Important Number Relationships (5)

A very puzzling feature exists as to the nature of results for the Riemann zeta function where s is a positive integer.

Now apart from the case where s = 1, the function has a finite value  However the form of this value differs distinctly as between even and odd values.

In the case where s = 2, ζ(2) = π²/6 and in fact where s is a positive even integer, the result can always be expressed in the form π^s/k, where k is a rational number.

So for example when s = 4, ζ(4) = π⁴/90; when s = 6,  ζ(6) = π⁶/945; when s = 8,
ζ(8) = π⁸/9450; when s = 10, ζ(10) = π¹⁰/93555. Now up to s = 10, k is an integer. However this pattern is broken when s = 12, with ζ(12) = 691π¹²/638512875.

The denominators associated with all these values are especially interesting in their own right for the following reasons.

When s is a power of 2, the denominator of ζ(s) is divisible by all the primes (and only those primes) to s + 1.

When s is not a power of 2, the denominator of ζ(s) is divisible by all the primes except 2 (and only those primes) to s + 1.

For example the denominator - as we have seen - for ζ(12)  is 638512875. Now 12 is not a power of 2. Therefore 638512875 is divisible by all the primes from 3 to 13 inclusive (and only those primes).

In fact 638512875 =  3⁶.5³.7².11.13. Some years ago, I made an extensive study of these denominators (based on the data at my disposal) and this appears very much to be a pattern that universally holds for all denominators of  ζ(s) where s is even.  

However, where s is a positive odd integer, no such relationship involving powers of π (multiplied by a rational number exists).  Indeed, no other convenient closed form relationship likewise appears to exist.

So therefore a fundamental difference characterises the behaviour of  ζ(s) for positive even and odd integers respectively.

Now we can perhaps throw some holistic light now on the reason for this situation.

In the last blog entry I showed how the probability that 2 numbers chosen at random possess no common factors, could be expressed in terms of the Type 2  aspect of the number system as where the fractional exponent (expressing an appropriate root of unity) consists of the 2 numbers involved.

So again in the example given there, if 2 numbers (chosen at random between 1 and 100) are 16 and 60, then this could be expressed in the Type 2 system as 1^16/60.

So in this case the fact that the two numbers 16 and 60 have common factors (i.e. proper factors), can be expressed in Type 2 terms by the fact that both the numerator and denominator of the fraction (expressing a root of 1) likewise contains a common factor.

And again as we have seen, the resulting root (– .1045 +.9945i)  provides an indirect quantitative means of expressing the notion of the 16th member (of a group of 60). So the very point about this is that what we have here the harmonious reconciliation of both the ordinal and cardinal notion of number. And this entails the corresponding reconciliation of both circular and linear number notions. So the 16th root in question here represent a point on the unit circle that is connected from the centre of the circle by a unit line! So the 16th root (in ordinal terms) represents one of the 60 (cardinal) roots of 1.

Likewise the proper understanding of this relationship (entailing both quantitative and qualitative notions of number) entails a corresponding reconciliation of holistic (intuitive) and analytic (rational) understanding (that are circular and linear with respect to each other).


However if we now consider the probability of 3 numbers chosen at random containing no common factor, this cannot be expressed with respect to the Type 2 system. Whereas any two numbers could be used as a fractional index (of 1) representing one of the roots of 1, the remaining number could not be expressed in this manner.

So for example our 3 numbers chosen at random (again between 1 and 100) are 12, 25 and 71
we could express any two together as a fractional index (of 1) e.g. 12/25, 12/71 or 25/71. However in each case there will be one number left over (that does not represent a fractional root),

Therefore the pure circular/linear relationship that characterises each root (in both quantitative and qualitative terms) cannot therefore apply in this case.

Indeed long before any mathematical considerations, I had looked deeply into this issue from a pure holistic perspective (relating to "higher" contemplative stages of development).

Now normal development is characterised by a considerable degree of  linear (1-dimensional) understanding that allows for unambiguous dualistic distinctions with respect to phenomena.

So the first major stage of contemplative spiritual development culminates in 2-dimensional appreciation, where opposite external and internal polarities (which fundamentally condition all phenomenal understanding) are understood in nondual fashion as ultimately fully complementary with each other. Expressed in holistic mathematical terms, these two polarities are understood paradoxically as + 1 and –  1 with respect to each other (so that the signs can interchange) when interpreted in the conventional linear manner.

Now of course these same polarities necessarily condition all mathematical understanding and this is why the conclusions that apply in contemplative spiritual terms equally apply in a holistic mathematical context!

I then spent many years trying to come up with a neat holistic mathematical manner of expressing the next 3-dimensional stage  before eventually realising that no such way existed.

So I gradually came to accept that the even dimensional numbers represented the more passive stages, where a certain contemplative equilibrium (entailing the balanced use of both reason and intuition) prevailed. However the odd dimensional stages represented more active stages, where the previous equilibrium would temporaily break down, requiring a new period of conflict btween reason and intuition before a "higher" contemplative equilibrium could once again be restored.

In particular during the 3rd dimensional stage, I was aware of a marked discrepancy as between new holistic insights (formed during the previous contemplative stage) and my ability to manage practical duties at the conventional level (requiring linear reason).

So the key problem therefore during the 3rd stage was this mismatch as between (circular) intuition and (linear) reason. This was then eventually resolved to a degree at the 4th dimensional stage, where a much more refined experience of the interaction between fundamental poles took place.

So in holistic terms at the 4th dimensional stage, balance is once more restored as between external and internal polarities (as - relatively - positive and negative with respect to each other) and the crucial whole and part polarities (i.e. qualitative and quantitative), that are now understood in refined holistic mathematical terms as real and imaginary with respect to each other.

So though we are speaking here about the four roots of unity, they acquire in this development context an entirely distinctive holistic meaning! And this is potentially true with respect to every mathematical symbol and relationship!


So when we now look at the Riemann zeta function, we can see the same mismatch with respect to values of s representing the odd integers.

Whereas linear and circular aspects (in both quantitative and qualitative terms) are successfully integrated with each other (where s is even) a certain displacement is in evidence (where s is odd). So the rational numbers are transferred to the LHS of the function (associated with negative values for s) whereas the irrational aspect remains on the RHS.

So the values of the Riemann zeta function (where s are positive odd integers) are indeed irrational but not in a perfect manner as represented by the transcendental number π. So no displacement of the rational aspect takes place for even values of s, which is indicated by the fact that the value of ζ(s) for all  negative even integer values of s = 0.

There is yet another simple way of appreciating the difference as between the value of ζ(s) for positive even and odd integers respectively.

When one obtains an even number of roots of 1, a direct complementarity is in evidence where each of one half of the roots can be exactly matched in each case by their negative counterpart (in the other half).

However where an odd number of roots is concerned, such complementarity does not exist, with one of the roots (i.e. 1)  standing somewhat apart, while the remaining roots exist as complex conjugates of each other.


Now if we briefly look at the probability that 4 numbers chosen at random will not contain a common (proper) factor, this will be given as 1/ζ(4) = 90/π⁴ = .9239...

We can therefore formulate this in Type 2 terms, this time requiring fractional numbers for 2 roots.

So if for example (again choosing at random from 1 - 100) we get for example the four numbers, 7, 31, 43 and 78, we could express these in Type 2 terms as 1^7/31  and 1^43/78. So the issue now depends on whether both numerator and denominator of both 7/31 and 43/78 contain a common factor.

Alternatively expressed this implies the probability that the product of the four numbers (i.e. 7 * 31 * 43 * 78) is not divisible by a prime factor (raised to the power of 4).

Riemann Zeta Function: Important Number Relationships (4)

So far, in the last 3 blog entries in this series, I have been adopting the conventional (Type 1) approach, which concurs with the standard analytical quantitative interpretation of number.

So even though number - depending on context - continually switches as between both a particle (analytic) and wave (holistic) identity, no recognition of this takes place in the conventional approach.

Now the basic issue - as I have continually reiterated in my blog entries - relates to the distinction as between the notions of number independence and number interdependence respectively.

Therefore, once again, when a prime such as 3 is employed in conventional mathematical terms, it is interpreted with respect to its independent quantitative status in a cardinal manner, i.e. where its sub-units 1 + 1 + 1 are viewed in a homogeneous impersonal fashion as - literally - devoid of qualitative characteristics.

This treatment then implicitly concurs with the number raised to the default dimension of 1, i.e. as a number defined on the real number line.

This likewise concurs with a linear rational mode of interpretation (based on the making of one-way unambiguous sequential logical connections). And again this is what I define as the Type 1 aspect of the number system.

So in Type 1 terms, 3 is defined as 3¹.

However as well as independence, we should equally recognise the interdependence of all numbers which - strictly - is of a qualitative (rather than quantitative) nature.

So if we return to our example of the prime number "3", there is equally a sense in which its sub-units can be understood as fully interdependent with each other. Now in direct terms, such recognition occurs in an intuitive holistic manner. However indirectly such holistic recognition (of number interdependence) enables one to then make ordinal distinctions at the linear level of understanding.

So effectively, the true intuitive holistic recognition of number is inevitably reduced in a merely quantitative rational manner at the conventional level of mathematical understanding.

And unfortunately such gross reductionism then pervades the understanding of every mathematical relationship in conventional (Type 1) terms.


Therefore, the proper recognition of the uniquely qualitative nature of number requires a distinctive complementary Type 2 approach (where it is defined in a circular manner).

Here every number is inversely defined with respect to a standard base of 1, which then can be raised to all the natural numbers (as powers or exponents representing dimensions).

So in Type 2 terms,  3 is defined as 1³. Now strictly this refers to the intuitive - rather than the rational - recognition of 3, as the quality of "threeness" (where the interdependence of its sub-units with each other are recognised). Of course as all numbers have both analytic and holistic interpretations, 3 as a number representing a dimensional power i.e. 1³, equally can be given an analytic meaning i.e. as a cube with common side of 1 unit! However it is the holistic meaning of "3" that I am referring to here in this context.

We can then indirectly express such holistic understanding in a rational manner (i.e. in 1-dimensional terms) by obtaining the 3 roots of 1 (i.e. 1^1/3, 1^2/3 and 1^3/3 respectively) where they now appear as circular or paradoxical to rational understanding.
The fundamental significance of these number conversions - which are completely missed from the conventional mathematical perspective - is that they then provide a unique means of indirectly converting the ordinal notions of 1st, 2nd and 3rd (in the context of 3 members) in a quantitative manner.

Well, strictly in fact one of these  results, i.e. 1^3/3 is not unique and reduces to 1! And this is always the case in relation to the last of the n roots of any number. And it is this reduced notion of ordinal meaning that thereby defines the conventional mathematical approach.

So in cardinal terms 3 = 1 + 1 + 1. Then in ordinal terms 3 = 1st + 2nd + 3rd.

However conventionally, 1st, 2nd and 3rd are  implicitly identified with the last roots of 1(1^1/1) of 2 (1^2/2) and 3 (1^3/3) respectively where they reduce down to 1 in each case.

Therefore from this perspective 1st + 2nd + 3rd = 1 + 1 + 1 (so that ordinal meaning is thereby successfully reduced in the standard quantitative cardinal manner!).

However when one recognises the true distinctive nature of the Type 2 aspect of the number system, the mathematical world of number is turned completely on its head.

Thus, from the Type 1 perspective, 3, as a prime, is unambiguously viewed as an independent "building block" of the natural number system (in quantitative terms).

However from the Type 2 perspective, 3 as a prime is already uniquely defined by its distinctive ordinal natural number members (1st, 2nd and 3rd) in qualitative terms. And these members are fully interdependent with each other in a merely relative manner (depending on context).

So again from the quantitative internal perspective, each prime is viewed in quantitative terms as an individual "unit of linear independence" (with respect to the natural number system).

Then from the corresponding qualitative perspective, each prime is now viewed as a collective "group of circular interdependence" with respect to the natural number system.

Therefore in order to embrace these complementary features of the number system, we must now conceive of the prime in dynamic relative terms based on the two-way interaction of both its quantitative (cardinal) and qualitative (ordinal) aspects. And this equally applies externally to the relationship as between the primes and the natural number system as a whole

Ultimately the number system is characterised by an incredible two-way synchronicity with respect to both its quantitative and qualitative aspects (internally and externally). And the true appreciation of this fundamental fact requires the very refined marriage of both analytic and holistic type understanding (where neither aspect is reduced in terms of the other).

Clearly however, the standard approach of interpreting the number system in an absolute quantitative manner greatly distorts proper understanding of its true nature.


We will now return to the subject matter of the past few blog entries in an attempt to provide a richer appreciation of the true dynamics involved.

I had earlier attempted to do this in "Another Interesting Relationship" and "Further Investigation"  on my companion "Spectrum of Mathematics" blog.

I then recognised subsequently that the π relationship I suggested with respect to the circular number system (based on the roots of 1) was in error.

So just to recap! We have now established (with respect to the Type 1 aspect of the number system) that the probability a number chosen at random will not contain s or more prime factors is 1/ζ(s).

So in the important case where s = 2, the probability that such a number will not contain 2 or more factors = 1/ζ2) = 6/π².
This equally can be expressed as the probability that a number chosen at random is square-free or alternatively the probability that 2 numbers chosen at random will contain no common (proper) factor!

With respect to the Type 2 circular system, the n roots of every number comprise a distinct part of the overall system. Now again the primes are unique in this respect in that the n roots of 1 for every prime (apart from the default root of 1) cannot be repeated for any other prime.

Thus as I expressed it in the earlier entry, the fractions expressing the roots for such prime numbers are irreducible.

However with the roots of the composite numbers, some fractions will be reducible and other factors non-reducible.
So my contention was that that this would be governed by the same relationship (as in Type 1 terms).

Therefore we can now state, the probability that the fractional value chosen at random (associated with the roots of 1 in Type 2 terms) is irreducible, is the same as the probability that a number chosen at random (in Type 1 terms) will not contain 2 or more factors.

So once again this probability is given as 1/ζ2) = 6/π².

In fact, given that we have already established that this equally expresses the probability that 2 numbers chosen at random (in Type 1 terms) contain no proper factors, the result must necessarily hold.

The reason for this is as follows. Any two numbers chosen at random can be expressed as the numerator and denominator respectively with respect to a fractional value in the Type 2 system.

For example let us choose two numbers (in Type 1 terms) ar random from 1 to 100. Imagine we obtain 16 and 60! Then 16/60 can be expressed in Type 2 terms as the fractional value associated with the 16th root of the 60 roots of 1.

So the issue in Type 1 terms as to whether these contain common factors, amounts in Type 2 terms as to whether both the numerator and denominator of the fraction involved contain common factors. 

And as 16 and 60 contain common factors, then equally the numerator and denominator of 16/60 contain common factors!.

However though we are confirming the same numerical result here in both cases, crucially it relates to two different interpretations of number!

In other words if the Type 1 result relates to the particle aspect of number interpretation - then - relatively - The Type 2 results relates to the wave aspect!

In fact this is very revealing in another context.

As we have seen the quantitative result (in both cases) = 6/π² .


Now when one looks at number in its proper dynamic (Type 3 context), which incorporates Type 1 and Type 2 aspects as interacting partners, this entails a linear frame of reference for the Type 1 aspect and a circular frame of reference for the Type 2 aspect of the number system respectively. 

Now the important symbol π, expresses in quantitative terms the (pure) relationship as between a (circular) circumference and its (line) diameter.

Equally in holistic terms, it expresses the pure relationship as between intuition (i.e. that appears  circular or paradoxical when indirectly expressed in a conventional manner) and (linear) rational type understanding.

Therefore the deeper holistic (i.e. qualitative) appreciation as to why these two relationships involve a simple relationship (involving π) is because the proper understanding of both relationships entails the coherent incorporation of both Type 1 (linear) analytic and Type 2 (circular) holistic understanding, entailing  notions of quantitative independence and qualitative interdependence respectively.

In fact we can even provide a rationale as to why the square of π is involved in the relationship.

From the Type 1 perspective,  6/π² expresses the probability that the product of two numbers (chosen at random) are square-free.

Then in Type 2 terms it expresses the probability that the product of the two numbers (representing numerator and denominator respectively) are likewise square-free. As we have now switched from a linear (Type 1) to a circular (Type 2) frame of reference, the dynamic explanation therefore entails the integrated relationship of circular and linear type aspects of the number system!

Therefore the richer interpretation of these simple results based on the Riemann zeta function, entails that both Type 1 (quantitative) and Type 2 (qualitative) aspects of the number system be incorporated with each other in a dynamic complementary manner.

Expressed more generally, complete understanding of the Riemann zeta functions entails at least 4 separate stages (two of which are only presently recognised in conventional mathematical terms).

1. The analytical understanding of the function that is confined to real quantitative values that are positive, which corresponds with the Euler zeta function. (Now I am of course aware of the more specialised mathematical interpretation of "analytic" in the context of the Riemann function. However I am using analytical here in a more general sense as relating to the conventional quantitative interpretation of number, which thereby includes the narrower specialised mathematical meaning of "analytic").

2. We then extend analytical understanding to complex quantitative interpretation (both positive and negative) which corresponds with present understanding of the Riemann zeta function.

However, though this does indeed enable significant advances to be made with respect to the quantitative relationship of the primes to the natural numbers, it is still crucially limited (and indeed distorted) by the attempt to understand number in a merely quantitative absolute manner.

3. The next stage in understanding entails the recognition that associated with every analytic (quantitative) interpretation with respect to the Riemann zeta function is a corresponding unrecognised holistic (qualitative) interpretation.

Though recognition of this holistic aspect did at once co-exist with the analytic, especially with the Pythagoreans  (although in a somewhat undeveloped manner) subsequently it has been completely discarded - at least in formal terms - from accepted mathematical understanding.

So it would be very difficult, I imagine, for most practising mathematicians to form any realistic conception of what this holistic aspect means.

I will attempt here however to give just one brief example - which if properly followed - will be seen to have immense repercussions for the misguided quest to seek a proof of the Riemann Hypothesis.

As is well known, the one value (in accepted Type 1 terms) for which the Riemann zeta function is undefined is where s (representing a common dimensional power with respect to the terms of the  function) = 1.

Now this - using my terminology - represents the accepted analytic interpretation (where number relationships are interpreted in an absolute quantitative manner).

The corresponding holistic explanation (in Type 2 terms)  again states that the function is undefined where s = 1. However 1 in this holistic context takes on the qualitative meaning of rational linear interpretation (which in fact defines the conventional mathematical approach).

So, remarkably what this entails, is that strictly it is not possible to properly understand the Riemann zeta function (i.e. it remains undefined) in an absolute 1-dimensional rational manner.

In other words, the true nature of the number system is inherently dynamic and interactive, entailing the two-way relative interaction of both quantitative and qualitative type interpretation.

And we can only give meaning to the mapping of values through the functional equation connecting positive values for s on the RHS with negative values for (1 – s) on the LHS, through recognition of the holistic aspect.

For example the conventional interpretation of the series 1 + 2 + 3 + ....  is that the sum continually gets larger and ultimately diverges (to infinity).

However according to the Riemann zeta function (where s = – 1) the sum of this series = – 1/12.

So at the very least, we are faced with a major task of interpretation in explaining how two completely different results can be apparently given for the same series!

And of course the real problem is that whereas number results concur with standard analytic interpretation on the RHS,  they then - in relative terms - concur with holistic interpretation on the LHS.

So the real purpose of the Riemann functional equation - when suitably interpreted - is that it shows how analytic can be mapped with holistic (and holistic with analytic) values with respect to both sides of the function. However again this entails understanding number as the two-way interaction of both quantitative (Type 1) and qualitative (Type 2) meaning.

So the very nature of the Riemann Hypothesis - when suitably understood - transcends conventional mathematical interpretation.

The key issue relates to the condition for the mutual identity of both the quantitative (analytic) and qualitative (holistic) type notions of number. And the Riemann Hypothesis amounts to a precise statement of this requirement!

However clearly it is futile attempting to prove this condition when the qualitative (holistic) aspect is not even recognised in formal mathematical terms. Expressed in an equivalent manner, the consistency of qualitative with the quantitative interpretation of mathematical symbols is already assumed in the axioms used to obtain mathematical proof! So these axioms therefore cannot be meaningfully used to obtain the proof (or disproof) of the Riemann Hypothesis.

So the fundamental revolution that is greatly required in Mathematics could eventually be slowly precipitated through the continued failure to prove the Hypothesis, eventually awakening mathematicians to the fact that there are indeed crucially important dimensions to the understanding of number that have hitherto been totally suppressed with respect to conventional interpretation. 

4. At the most advanced stage of understanding, the Riemann zeta function -  at least in terms of what I can envisage - both Type 1 (analytic) and Type 2 (holistic) aspects are dynamically incorporated in an integrated harmonious fashion. This is what I refer to as Type 3 understanding, or alternatively radial mathematical understanding.

Even within this category, I would refer to three - relatively - distinct subcategories.

With Type 3 (a) the emphasis is primarily on creative analytic type developments (within the context of established holistic appreciation).
With Type 3 (b) the emphasis is primarily on holistic type interpretation (against a background of established analytic appreciation).
I would see my own recent approach - in a most preliminary fashion - as representative of this subtype. 
In other words, though I necessarily grew up - even to the extent of studying Mathematics at University - with the accepted analytic approach, my ability is primarily with respect to the (unrecognised) holistic dimension. So  in this context, I would see this contribution with respect  to the Riemann zeta function as one of radical re-interpretation, where the very nature of the number system is now interpreted in a completely new light.
Type 3 (c) would then represent the pinnacle of mathematical understanding, where one would be equally gifted with respect to both analytic and holistic type appreciation enabling a contribution to Mathematics both highly creative (holistically) and immensely productive (in analytic terms). 

So a proper understanding of the Riemann zeta function (and with it the number system) will require eventual progression to stages 3 and 4. Thus the  holistic aspect of mathematical understanding firstly will need to be recognised as equally important - though utterly distinct - from the analytic. Then when this neglected aspect undergoes appropriate specialisation, both the analytic and holistic aspects will finally be required to be coherently integrated with each other in a harmonious interactive manner. 

Riemann Zeta Function: Important Number Relationships (3)

I mentioned before in "Fascinating Connections" on my companion "Spectrum of Mathematics" how the extended Fibonacci sequence can be used to closely approximate the Riemann zeta function for positive real values of s.

In like manner therefore we can use this extended sequence to provide increasingly good approximations for the value of 1/ζ(s) where s is a positive integer, which as we saw yesterday plays an important role with respect to the distribution of the primes.

So once again in general terms 1/ζ(s) represents the probability that a number, chosen at random will not contain s or more repeating prime factors.
Likewise as we have seen, it expresses the probability that the number will not be divisible by a prime factor raised to the power of s or alternatively the probability that s numbers chosen at random will not contain a common (proper) factor.

Now the famed Fibonacci sequence relates to the equation,

x² – x – 1 = 0 with phi (the golden ratio = 1.618... the positive real valued solution for x to this equation).

Alternatively, the ratio can be estimated by starting with the digits  0, 1 and then successively adding the last digit (in a 2-digit sequence) to the previous digit to thereby obtain 

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, .....

Then the ratio (i.e. phi) can be approximated to increasing degrees of accuracy by dividing the last number in the sequence by the previous number.

Therefore in this case we get 144/89 = 1.617977... which is already a very good approximation to the true value 1.618033....

Now in general terms - what I refer to as - the extended Fibonacci sequence relates to the equation

x^s – x ^{s – 1} – ...... – x – 1 = 0.

In every case for s ≥ 1, a positive real-valued solution for x will arise.

Though we could algebraically obtain a solution where s = 3 and 4, the simplest approach (which can be extended to larger values of s) is by an extension of the method we used to obtain the terms in the Fibonacci (2-digit) case.

So for example in the 3-digit case (corresponding to x³ – x² – x – 1 = 0) we start with the 3 digits

0, 0, 1,  and now add - starting with the last in the sequence - the 3 previous digits.

In this way we obtain the following sequence,

0, 0, 1, 1, 2, 4, 7, 13, 24, 44, 81, 149,....

The estimate for x (as the positive real valued solution for x) here = 149/81 = 1.839506... which is very close to the true estimate 1.839286....).

The basic contention is that x – 1 (where x is the solution to the general equation,

 x^s – x ^{s – 1} – ...... – x – 1 = 0) offers an excellent approximation for the corresponding value of 

1/ζ(s).

For example, when s = 3, as the relevant solution to the general equation for x = 1.839 (approx), then x – 1 = .839.

And the corresponding value of 1/ζ(s) for s = 3 is .832 (approx). So the Fibonacci type estimate  is already very close and steadily improves in accuracy for larger values of s.

Once again as 1/ζ(s) represents the probability that a number, chosen at random will not contain s (or more) repeating prime factors, then 1 – 1/ζ(s) therefore represents the probability that the number will contain s (or more) repeating factors.

Now .the ratio of the probability that a number will contain s (or more) repeating factors to the probability that it will not contain these factors = {1 – 1/ζ(s)}/1/ζ(s) = ζ(s) – 1.

And ∑{ζ(s) – 1} = 1, where s = 2, 3, 4, 5,...... .

Thus  the ratio of the probabilities that a number will contain a certain number of (proper) factors (or more) to the probability that it will not contain those factors = 1 (when summed for s ≥ 2).

Riemann Zeta Function: Important Number Relationships (2)

We saw yesterday the importance of the Riemann zeta function ζ(s), where s = 2 for the distribution of the prime factors of the various members of the number system.

So the probability that a number chosen at random will be composed of non-repeating prime factors = 1/ζ(2) = 6/π² = .6079 (approx). Alternatively, the probability that it will be composed of repeating prime factors (2 or more) = 1 – 1/ζ(2) = 1 – 6/π²  = .3921 (approx).
 

1/ζ(2) could also be stated as the probability that a number is square-free i.e. not divisible by the power of 2 (or higher) of any prime.

Equally it could be stated - at least in conventional (Type 1) mathematical terms - as the probability that two numbers chosen at random will not contain a common (proper) factor.

Therefore 1 – 1/ζ(2) represents the probability that a number is not square-free i.e. divisible by a prime raised to the power of 2 (or higher). Equally it represents the probability that 2 numbers chosen at random will contain a common factor. 


However, though in this context, 1/ζ(s) for s = 2 has the most general relevance for the number system, we can however give an important meaning to 1/ζ(s) where s is any other positive integer.

In this regard, it is interesting to probe the significance of the limiting case for 1/ζ(s), where s = 1.

Now clearly in this case, ζ(1) in Type 1 terms → ∞; therefore 1/ζ(s) = 0  and 1 – 1/ζ(1) = 1.  So we could interpret this as the probability that a number will be composed of 1 factor (or more), which of course is necessarily the case. Alternatively, this represents the probability that a number is divisible by the power of 1 (or higher) of some prime or alternatively the chance that a single number chosen at random will contain one factor (in common with itself) which again is necessarily the case.

Then for example when s = 3, 1 – 1/ζ(3) expresses the probability that a number chosen at random will be composed of 3 (or more) repeating prime factors. This could also be expressed as the probability that a number is divisible by a prime (raised to the power of 3 or higher) or alternatively the probability that 3 numbers chosen at random will contain a common factor.

Now ζ(3) = 1.20205693... and 1/ζ(3) = .8319 (approx).

Therefore 1 – 1/ζ(3) = .1681 (approx).

So the probability that a number chosen at random will contain 3 (or more) repeating factors is roughly 1/6. Alternatively we could express this result as the probability that a number is divisible by a prime raised to the power of 3 (or higher) or the probability that 3 numbers chosen at random will contain a common factor.

In more general terms, 1 – 1/ζ(s) expresses the probability that a number chosen at random will contain s (or more) repeating factors. Alternatively it expresses the probability that a number is divisible by a prime raised to the power of s (or higher) or the probability that s numbers chosen at random will contain a common factor.


We can use these results to obtain a more precise knowledge of the prime factor composition of the members of the number system.

For example 1 – 1/ζ(2) represents the probability that a number will contain 2 (or more) prime factors that repeat, while 1 – 1/ζ(3) represents the probability that 3 (or more) prime factors that repeat.

Therefore to obtain the probability that a number will contain at most 2 prime factors that repeat we subtract the latter from the former i.e.  1 – 1/ζ(2)  – {1 – 1/ζ(3)} = 1/ζ(3) – 1/ζ(2) = .8319 – .6079 = .224.

In more general terms we can therefore state that the probability that a number will contain at most s prime factors that repeat  = 1/ζ(s + 1) – 1/ζ(s).

To illustrate this, we will now estimate the total of those numbers (up to 100) which contain at most 3 repeating prime factors.

The probability (for the number system as a whole) is given as 1/ζ(4) – 1/ζ(3) = .9239 – 8319 = .092.

This would suggest therefore that we should find (up to 100) approximately 9 numbers that contain at most 3 repeating prime factors.

In fact the full list is given below:

8 =   2 * 2 * 2 

24 = 3 * 2 * 2 * 2

27 = 3 * 3 * 3

40 = 5 * 2 * 2 * 2

54 = 3 * 3 * 3 * 2

56 = 7 * 2 * 2 * 2

72 = 3 * 3 * 2 * 2 * 2

88 = 11 * 2 * 2 * 2

So there are 8 such numbers (with 3 or more repeating prime factors) up to 100 which is close to our prediction of 9.

When we carry on further up to 200, a further 9 numbers contain at most 3 repeating prime factors! So the early sample evidence is very much in line with the predicted result.

In fact there is a remarkable stability with respect to such results throughout the number system with variations from predicted results quite small.

Riemann Zeta Function: Important Number Relationships (1)

I wish here to correct an earlier error on my “Spectrum of Mathematics” blog with respect to “Remarkable Features of the Number System“ (1 – 4) which was later followed up in “Another Interesting Relationship” and “Further Investigation” and also on the present “Riemann Hypothesis” blog with respect to “Holistic Synchronicity” (1 – 3).

In those entries I suggested that a certain constant relationship (based on π) would hold with respect to the number system as a whole regarding the distribution of those numbers with non-repeating prime factors and repeating prime factors - or structures as I referred to them - respectively.

So based on some initial empirical investigation at various intervals of the number system, I concluded then that the average frequency of those numbers with non-repeating prime factors would be π/(π + 2) = .611 (approx) with the average frequency of the remaining numbers with repeating prime factors 2/(π + 2) = .389 (approx).

However when reading about - what are referred to in the literature as - square-free numbers, I realised that this in fact represented an alternative way of stating my position.

Thus the average frequency of square-free numbers (i.e. numbers that are not divisible by prime factors raised to the power of 2 or higher) would by definition represent those numbers with non-repeating prime factors.

And it is well-known that the average frequency of such numbers (or alternatively the probability of obtaining a square-free number) = 6/π², which of course is the corresponding value of the Riemann zeta function for 1/ζ(s) where s = 2.

Now 1/ζ(2) = 6/π² = .608 (approx) which is very close to π/(π + 2) = .611 (approx).

In fact, in reaching my earlier erroneous conclusion, I had initially considered 6/π² the most likely estimate. However as the sample values that I took at various intervals (to estimate the frequency of numbers with non repeating prime factors) repeatedly averaged out very close to .611, I then changed to an alternative π estimate.

So now in retrospect, it is apparent that the consistency of my estimates represented in fact sample bias over the number intervals that I chose.

In any case it is quite easy to establish the correct result (without resort to empirical data).

The probability that a number is divisible by the square of a prime p = 1/p².

So for example in the simplest case where p = 2, we would expect 1 in 4 numbers to be divisible by 2².

Therefore the probability that a number is square-free with respect to a given prime (i.e. not divisible by the square of a particular prime) = 1 – 1/p².  

However as well as considering divisibility by the square of 2, we equally have to consider all the other primes i.e. 3, 5, 7, ….

So therefore the probability that a number is generally square-free with respect to any prime = ∏(1 – 1/p²)  = 1/ζ(2) = 6/π².And as we have seen, this equally represents the probability that a number is composed of non-repeating prime factors, or alternatively, the average frequency of numbers with non-repeating prime factors.


And this result can be easily generalised as the probability that a number cannot be divided by a prime (raised to the power of n) = 1 – 1/pⁿ.  And then when we allow for divisibility by all the primes we get ∏(1 – 1/pⁿ) = 1/ζ(n).  

Then returning to the case for s = 2, this then entails that the probability that a number is not square-free i.e. is composed of at least one repeating prime factor = 1 – 6/π².

However there is yet another way of stating this important relationship.

The probability that a number is square-free, in fact represents the probability that any two randomly chosen numbers will contain no common factors.

And once again this probability is given as 1/ζ(2) = 6/π².

Let us consider for example the two numbers 6 and 8. These contain 2 as a common prime factor. Alternatively we could express this situation as the fact that the product of 6 and 8 (= 48) is necessarily divisible by the square of 2.

Therefore if two numbers chosen at random contain common prime factors then the product of those numbers cannot be square-free.

However, clearly if the two numbers do not have common prime factors, then the product of these numbers is square-free.