The Riemann transformation formula establishes an important link as between values of the zeta function for s > o on the RHS and corresponding values for 1 - s on the LHS of the equation. Now crucially from the conventional linear perspective, the zeta function will only converge for finite values of s > 1.

Therefore no adequate explanation can be given in linear terms for values of the zeta function where s < 1.

So the Riemann Zeta Function can be neatly subdivided for all real values of s as follows:

(i) as representing the standard reduced linear interpretation (directly in accordance with the dimension 1) for values of s > 1.

ii) as representing the alternative circular interpretation (directly in accordance with the dimensional number in question) in corresponding qualitative terms for s < 0.

Interestingly then the values in the critical range for 0 < s < 1 represent a hybrid mix of both quantitative and qualitative aspects. The condition therefore for equality with respect to both sides (with reference to the non-trivial zeros where both sides of the equation = o) is that s = 1 - s = .5.

So this condition representing the famous Riemann Hypothesis where for all non-trivial zeros, the value of the real part = .5, is required so as to obtain an exact correspondence as between both quantitative and qualitative type interpretation.

We can say therefore that when this condition is realised that there is no longer any gap as between the quantitative nature of prime number reality and our corresponding qualitative interpretations of such reality. This would equally imply that the quantitative nature of individual prime numbers can then be perfectly reconciled with the overall qualitative holistic nature of their distribution (among the natural numbers).

And once again because the Riemann Hypothesis - when correctly interpreted - points to this essential requirement for reconciliation of both quantitative and qualitative aspects with respect to prime number behaviour (and thereby by extension all number behaviour), it cannot be proved (or disproved) with respect to just one aspect i.e. the axioms pertaining to the conventional quantitative linear approach.

What the Riemann Hypothesis is directly implying in fact is that there are two equally important aspects to mathematical understanding i.e. quantitative and qualitative. However in the present mathematical approach (which has dominated understanding now for several milennia) only one of these aspects is formally recognised i.e. the quantitative.

There is I believe however a very simple way of expressing the relevance of the Riemann Hypothesis.

If we draw a circle and insert its line diameter, the point at the centre of the circle is equally the point at the centre of the line diameter. So in this sense both the line and the circle are identical at this mid-way point. So if we identify the line diameter as 1, the midpoint occurs at .5.

In similar qualitative terms the reconciliation of both linear (quantitative) and circular (qualitative) interpretation occurs at the same point. So the midpoint in this context represents the situation where opposite polarities of experience are perfectly balanced.

Thus maintainence of the most refined interaction possible as between (linear) reason and (circular) intuition, requires that these opposite polarities of understanding (which necessarily underline all phenomenal understanding of reality) be kept in perfect balance. In this way, one can temporarily separate and differentiate opposites (e.g. external objective phenomena and internal mental constructs) with respect to either pole in a refined rational manner while immediately seeing from an integral holistic perspective that both poles are complementary (and ultimately identical). In this way quantitative rational interpretation and qualitative intuitive appreciation can dynamically approximate the situation where they can then serve each other perfectly. However while approaching such an approximation a continual correction mechanism is required whereby unconscious i.e. imaginary projections (representing temporary imbalance as between the discrete nature of reason and continuous nature of intuition) are constantly emitted.

And the key to rapid adjustment here, is that like virtual particles in physics, these temporary imaginary projections should occur in pairs (whereby they can quickly cancel each other out).

This equally implies to the very nature of prime numbers, entailing an identical similar process where both the individual quantitative nature of each prime number can be kept in perfect balance with the qualitative holistic nature of the distribution of primes (among the natural numbers). And once again temporary imbalances as between the discrete nature of individual primes and the continuous nature of their general distribution are represented through appropriate imaginary dimensional numbers added to the real part of s (that occur in pairs). So prime number behaviour - properly understood - represents the interaction of two logical processes (linear and circular) that are kept perfectly in balance through the constant adjustments brought about through these imaginary dimensional number pairs.

So once again the quantitative nature of prime numbers cannot be ultimately distinguished from the corresponding qualitative nature (by which they are interpreted).

And this is the key message of the Riemann Hypothesis!

I feel the same, but I want to thank you that you formalised my vague thoughts!

ReplyDeleteI also want to add that although an exact pattern of the primes (at last!) is revealed through using complex numbers it is the real part of them that shows the pattern, the regularity. This goes to show leaps of thought are needed to tame these numbers. We had to bring negative numbers and then their square roots to find a constant for the prime numbers!