So far we have confined ourselves with solutions with respect to the finite expression of the Zeta 2 Function,
ζ2(s) = 1 + s1 + s2 + s3 +….. + st – 1 (with t prime) = 0
Now this can be extended in an (conventional) infinite manner provided we only take groups of t terms.
So for example in the simplest case where t = 2 (where the finite solution is s = – 1) , the infinite expression,
ζ2(s) = 1 + s1 + s2 + s3 +….. = 0,
will still have the solution – 1, provided that we maintain an even number of terms by extending in pairs.
However the problem then arises as to what the value of the expression ζ2(s) will be when we remove this restriction that terms be taken in groups of t.
Again in the simplest case where s = – 1, we generate the alternating series,
ζ2(s) = 1 – 1 + 1 – 1 + 1 – 1 +......
There are only two possible values to this infinite series.
If we take an odd number of terms, ζ2(s) = 1;
however if we take an even number of terms ζ2(s) = 0.
Since the probability of an even or odd number of terms is equal, then we could postulate the probable value of the series as the mean of both values = 1/2.
Fascinatingly 1/(1 – s) = 1 + s1 + s2 + s3 +…..
and when we put in the value s = – 1 in the expression we get
1/2 = – 1 + 1 – 1 + 1 – 1 +......
So the value here obtained by the formula (though non-intuitive in linear terms) provides the same answer as we earlier achieved through probabilistic reasoning.
This would therefore strongly suggest that the value 1/2 is a numerical value representing a probability.
We can put it even more strongly by saying that this very notion of probability is inherent in the Riemann Hypothesis, so that our standard probabilistic notions only hold on the basis that the Riemann Hypothesis likewise holds.
When for example we say that the probability of getting"Heads" in tossing an unbiased coin = 1/2, there is a unavoidable element of circularity in this definition.
In fact the notion of probability here entails the assumption of a consistent relationship as between finite and infinite notions.
Clearly when we toss a finite number of coins the actual frequency of the number of Heads is likely to deviate from 1/2.
Then the assumption that the result will tend ever closer to 1/2 (as we keep increasing the number of tosses), in itself requires the initial assumption that "Heads" and "Tails" are equally likely!
Thus what really lies behind the notion of probability is the fundamental issue of the consistency of finite and infinite notions (which are qualitatively distinct).
So when we maintain that the probability of obtaining "Heads" = 1/2, we initially intuit this in infinite terms (as potentially applying in all cases).
Then we attempt to maintain consistency as between this infinite result (that potentially holds) and finite behaviour (that applies in actual cases). So underlying our conclusion, that in repeated finite trials using an unbiased coin, the result will tend ever closer to the potential infinite result ( = 1/2) is the implicit assumption that finite and infinite behaviour do indeed correspond with each other.
Thus the finite (actual) behaviour of repeated finite trials of an unbiased coin is ultimately assumed to be fully consistent with its assumed infinite behaviour (in potential terms).
And indeed this would make sense. As I have stated previously linear are based on absolute notions. However circular (dimensional) notions are bases on relative notions.
However this truth can only be maintained in a relative approximate manner.
Indeed one could validly maintain that the assumption of the Riemann Hypothesis, that all the trivial non zeros (of the Zeta 1 Function) lie on the line with real part = 1/2, is exactly the same assumption that allows us to maintain that the probability of getting "Heads" with an unbiased coin = 1/2.
However this proposition cannot be proven in any absolute fashion as the finite behaviour of the coins can only approximate this assumed infinite behaviour in a relative manner.
So proof in conventional mathematical terms assumes an absolute (reduced) correspondence as between actual finite and potential infinite behaviour. Thus the truth of a general proposition e.g. the Pythagorean Theorem as potentially applying in all (infinite) cases is assumed to directly correspond with all (finite) cases in actual terms!
However with the simplest case of probability i.e. the probability of "Heads" or "Tails" on tossing an unbiased coin, we can see that the correspondence as between finite and infinite behaviour is merely of a relative approximate nature.
Though mathematicians may be loath to admit this - as it undermines all their basic assumptions regarding the nature of mathematical proof - the behaviour of the primes with respect to the natural numbers is likewise of a merely relative approximate nature, entailing the interaction as between infinite (qualitative) and finite (quantitative) notions.
Now it still remains a wonderful mystery how consistency is still preserved as between finite (analytic) and infinite (holistic) notions (when their distinctive nature is properly appreciated).
However this consistency is now necessarily of a probable rather than absolute nature. And again the Riemann Hypothesis establishes precisely the condition (with respect to the behaviour of the primes) for this consistency. So the relevance of 1/2 in this context (just like our chance of "Heads") is strictly of a probabilistic nature.
Again in returning to our formula,
1/(1 – s) = 1 + s1 + s2 + s3 +….. ,
when the value given in the formula seems non-intuitive from a conventional linear perspective, this implies that we must now switch to The Type 2 notion of number (which represents a dynamic interaction, with a merely probable numerical value).
This is a very significant point for - as we shall see - the value here of 1/2 ultimately is the basis for the value of the real part of the Zeta 1 zeros = 1/2, on which the Riemann Hypothesis centres.
The importance of this can be demonstrated for the Zeta 1 Function in providing a way for calculating its (non-intuitive) values, where s < 1.
Indeed I have used it in another blog entry to manually calculate the values for ζ1( – 1), ζ1( – 2) and ζ1( –3) respectively.
However an even more surprising factor is that when any of the possible root values are substituted for s in the infinite Zeta 2 Function, the same answer of 1/2 is obtained.
This needs some explaining so I will illustrate now with reference to the next prime number 3 (for its two non-trivial roots).
These two roots are – 1/2 + .866i and – 1/2 –.866i respectively.
Thus there are 3 possible options in terms of the number of terms in the series i.e. t, t + 1 and t + 2 respectively.
Now if with respect to the 1st possible value i.e. – 1/2 + .866i, we take the first 3 terms the sum of series = 0.
So for example s = – 1/2 – .866i and s2 = – 1/2 + 866i.
Therefore the sum of 1st 3 terms = 1 – 1/2 – .866i – 1/2 + 866i = 0. And this will repeat with succeeding groupings of 3 successive terms.
Then when we take t + 1 terms the sum = 1.
And when we take t + 2 terms, the sum = 1 – 1/2 + 866i. = 1/2 + .866i
However we can also calculate 3 possible values with respect to the other root, i.e. – 1/2 –.866i .
Once again the sum of first 3 successive terms = 1 – 1/2 –.866i – 1/2 +.866i = 0.
The sum of t + 1 terms = 1.
Finally the sum of t + 2 terms = 1 – 1/2 – .866i = 1/2 – .866i .
Thus is we now take the sum of the 6 possible values (using the two non-trivial roots in question) the total sum = 0 + 1 + 1/2 + .866i + 0 + 1 + 1/2 – .866i = 3.
Therefore as 6 values are included here the average mean value = 3/6 = 1/2.
Therefore the value of 1/2 once again represent the expected value (average mean value) over all the possible combination of terms with respect to corresponding non-trivial roots of t (where t is prime).
In fact there is another fascinating link here with the Zeta 1 Function.
As we have seen in our illustration for t = 3, the number of possible values = 3 * 2, i.e. t * (t – 1).
and earlier for t = 2, the number of possible values = 2 * 1.
However with respect to the natural number terms t, the frequency of primes = log t
Therefore if we use prime number groupings, the number of possible values = t * (log t – 1)
Then because these root values relate to circular - rather than linear - numbers when we now attempt to convert these to linear form by dividing t by
we obtain t/2π(log t/2π - 1) = t/2π.log t/2π - t/2π (which is the formula for calculation of the frequency of non-trivial zeros for the Zeta 1 Function).
Thus the trivial zeros for the Zeta 1 Function can thereby be seen as a linear expression (in imaginary number format) of the circularised version of the corresponding non-trivial zeros with respect to the Zeta 2 Function! And remember I have frequently expressed that the very significance of the imaginary notion in holistic mathematical terms is that it provides a manner of expressing notions that are inherently circular (paradoxical) in a linear format!
Remarkably therefore the two sets of zeros represent distinctive ways of expressing what ultimately represents the same identity i.e. differing perspectives on the same ultimate reality.
And these perspectives entail the dynamic interaction of both the quantitative (cardinal) and qualitative (ordinal) aspects of the number system, which can initially be viewed from two distinctive perspectives, but which ultimately are identical in a manner where experience of number approaches a pure ineffable state.