What has long interested me however is a certain unexpected property of the Zeta 2 Function, which then becomes enshrined in the Zeta 1 Function as its central feature.
Once again the Zeta 2 Function starts from the consideration of each prime as comprising a unique grouping of natural numbered objects (in ordinal terms).
So once again to illustrate, 3 is a prime which - by definition - is thereby compromised of 1st, 2nd and 3rd members.
Now when 3 is defined in the standard cardinal manner, it is comprised of homogeneous units in quantitative terms, i.e. 3 = 1 + 1 + 1.
This notion of 3 in Type 1 terms is then represented as 31. So 3 is here explicitly defined as a number quantity with respect to 1 (representing the 1-dimensional number line) which remains implicit.
However because quantitative and qualitative notions interact in a complementary manner, therefore to represent the corresponding qualitative notion of 3, it now explicitly is represented as a dimensional number that is defined with respect to an implicit base number that is 1.
What this entails in effect is that we cannot explicitly recognise qualitative ordinal distinctions (i.e. relationships between numbers) without implicitly recognising each number as an independent unit in quantitative terms.
So in the first case we are concentrating on the independent nature of the number 3 (in quantitative terms).
In the second case we are now concentrating on the interdependent nature of 3 (in qualitative terms).
But both notions in dynamic interactive terms are of a merely approximate nature.
In this way we can keep switching as between cardinal (quantitative) and ordinal (qualitative) notions of 3 in a relative manner.
The big issue then arises as to how one can convert - as it were - the qualitative notion of 3 (expressing the relationship between its individual members) in a quantitative manner.
And because the qualitative notion is literally in this context 3-dimensional, this entails taking the three roots of 1.
Put more accurately in requires taking the cube root of 11, 12, 13 respectively i.e. 11/3, 12/3 and 13/3
These 3 roots of 1 then uniquely express in a quantitative manner the notions of 1st, 2nd and 3rd (in the context of 3 members) .
So expressed as an equation, 1 = s3.
Now one solution (i.e. 1 = s) is trivial, as it is always one of the roots of unity. Thus dividing by
1 – s, we get 1 + s1 + s2 = 0.
The two solutions to this equation could thereby be referred to as the non-trivial zeros. In this case these would represent the 1st and 2nd members of a group of 3. The third member (of a group of 3) would be represented by the trivial solution . In other words when we already know the 1st and 2nd members, the 3rd member therefore can be unambiguously identified in an absolute type manner.
So generalising for all primes the Zeta 2 Function can be expressed as the solutions to the finite equation 1 + s1 + s2 + s3 +….. + st – 1 = 0 (where t is prime).
Then because of the unique relationships as between the primes and the natural numbers (where every natural number is composed of a unique combination of prime factors),
1 + s1 + s2 + s3 +….. + st – 1 = 0 (where t is a natural number).
Thus for any natural number t, we have t – 1 non trivial solutions. These represent in a quantitative manner the 1st, 2nd, 3rd,..... (t – 1) ordinal rankings with respect to t members of a group.
The issue the arises as to to what happens when this finite equation is extended in an infinite manner.
In other words what can we say about the value of the infinite Zeta 2 equation
1 + s1 + s2 + s3 +…. or alternatively s0 + s1 + s2 + s3 +… ?
This seems very interesting as its terms bear an inverse form to the Zeta 1 Function,
i.e. 1 – s + 2 – s + 3 – s + 4 – s + .....
Now returning to the infinite Zeta 2 expression, what we are now attempting to do is to estimate its value for each set of zeros that arise when this is finite.
Now the simplest case is for t = 2, where just one trivial zeros arises i.e. – 1.
Then substituting this value in infinite expression we get the alternating series
1 – 1 + 1 – 1 +........
Now clearly if the infinite series has an even number of terms its value = 0.
If however it has an odd number of terms, its value = 1.
Now because the probability of the series having an even or odd number of terms is equal i.e. 1/2 then we can say that its expected value = (0 + 1)/2 = .5.
And this in fact is the value that is customarily given for this series.
However initially, it might not seem at all obvious what the expected value of the series might be when a multiple set of non-trivial zeros arise (as in every case where t > 2 in the finite series).
So we will now explore the case where t = 3. This means that the two relevant zeros are – .5 + .866i and – .5 + .866i respectively (expressed correct to 3 decimal places).
Now when the first value is substituted in the infinite equation i.e. 1 + s1 + s2 + s3 +…., we get
1 – .5 + .866i – .5 – .866i, which then keeps recurring with every 3 terms.
Therefore if we assume that the infinite series is made up of multiples of 3 i.e. 3n terms, then the value of the infinite series = 0.
If however the infinite series comprises 3n + 1 terms, its value = 1.
Finally if the infinite series comprises 3n + 2 terms, then its value = .5 + .866i
Thus we have 3 possible values here!
However we must equally consider the value of the infinite series for the other zero!
So the first 3 terms (which will then continually repeat) are
1 – .5 – .866i – .5 + .866i .
Once more if the infinite series comprises 3n terms its value = 1.
If the series comprises 3n + 1 terms its value = 0.
Finally if the series has 3n + 2 terms its value = .5 –.866i.
Thus considering both possible zeros in this case (where t = 3) we have 3 * 2 ( = 6) possible values for the series.
Now on the assumption that all these outcomes are equally likely, the expected value of the infinite series = (1 + 0 + .5 + .866i + 1 + 0 + + .5 –. .866i )/2 = 3/6 = .5.
Now in more general terms the series for any finite value t will yield t * (t – 1) possible values for the infinite series.
And it is postulated here that the expected value in all cases = .5!
When looked on in the appropriate manner, this result is immensely revealing.
It is no accident that with the Riemann Hypothesis (in relation to the Zeta 1 function) that all non-trivial zeros are postulated to lie on the imaginary line through .5.
In fact the unexpected link with the much simpler Zeta 2 infinite function provides a remarkably simple way of expressing the true nature of the Riemann Hypothesis which is intimately tied up with the probabilistic nature of reality.
The very essence of probability is that it tries to bridge both finite and infinite realms (which are quite distinct from each other).
We could equally say that both finite and infinite realms relate to the quantitative and qualitative aspects respectively of the number system.
Now if we take the simple case of tossing an unbiased coin, we may well maintain that the probability of getting a H or a T is equally likely i.e. = .5.
However there is a subtle problem here. The postulate that both outcomes are equally likely strictly relates to a potential infinite order.
However when we empirically carry out trials where we repeatedly toss the coin, we are now dealing with the actual finite realm.
Now of course with a finite number of trials the number of H's and T''s recorded is unlikely to be equal. However the assumption that is made - which is strictly an act of faith - is that somehow the (actual) finite can eventually be successfully bridged with the (potential) infinite case.
So therefore if we were to keep increasing the number of tosses, we would be confident that the actual behaviour (of recorded tosses) would approximate ever more closely to the assumed potential behaviour in the infinite case (i.e. that both outcomes = .5).
Therefore we can look at the Riemann Hypothesis as the very condition that is required to justify the very assumptions that are made with respect to all probabilistic behaviour (which properly transcends rational behaviour).
Put even more simply the Riemann Hypothesis is required to properly underpin our assumption that repeated tosses of an unbiased coin will approximate ever more closely to an equal outcome of H's and T's .
Now as this assumption - which is properly an act of faith - entails a relationship as between both finite and infinite (or alternatively quantitative and qualitative) aspects of understanding, it cannot thereby be rationally proved (using the accepted axioms of Mathematics). These are based on merely reduced quantitative notions, which allow for no distinct holistic - as opposed to analytic - appreciation of mathematical symbols.
Thus in the end the true significance of the Riemann Hypothesis is that the very nature of the number system (and all Mathematics) is inherently dynamic, entailing a two-way interaction as between finite and infinite notions (i.e. quantitative and qualitative meaning).
In short the very nature of mathematical reality - when appropriately understood - is that it is inherently probabilistic!