Presuming the truth of the Riemann Hypothesis, all Zeta 1 (Riemann non-trivial) zeros come in pairs with the complex conjugate form 1/2 + it and 1/2 – it respectively.
So again the first pair of zeros are 1/2 + 14.134725i and 1/2 – 14.134725i respectively (correct to 6 decimal places).
In fact it is believed that the imaginary part of all zeros is of a transcendental nature.
So therefore each zero as a complex number, given the truth of the Riemann Hypothesis, contains a real part = 1/2 and an imaginary part of a transcendental nature.
In previous work, I spent some time exploring the precise holistic mathematical significance of all the number types (of which those belonging to the imaginary transcendental set are the most rarefied).
As we have seen, in this context, the significance of the imaginary notion is that it offers an indirect analytic means of conveying meaning that is directly of a holistic nature.
So the true nature of the interdependent relationship between the unique prime factors of the natural numbers is directly of a qualitative holistic nature. And the individual zeta zeros represent this relationship (which directly complement the opposite notion of the independent quantitative nature of the primes). However indirectly this relationship can then be expressed in an imaginary quantitative manner.
And just as all the real numbers are presumed to lie on a line (i.e. the number line), in like fashion, the zeta zeros - as the dynamic complementary expression of this line - are presumed therefore to lie on an imaginary line (which again is necessarily true if the Riemann Hypothesis holds).
So we can now perhaps begin to appreciate the true significance of the Riemann Hypothesis (which is completely overlooked in conventional mathematical terms).
Because of its limited absolute nature (where qualitative notions are reduced to quantitative) it is conventionally assumed as axiomatic that all real numbers lie on the same number line!
However, when one properly recognises that numbers necessarily possess a qualitative aspect (of relational interdependence) as well as a quantitative aspect (of individual independence) the number system, in dynamic terms, is thereby relative in nature, whereby quantitative notions of relative independence and qualitative notions of relative interdependence ceaselessly interact with each other in a bi-directional manner.
So the key issue that arises, in this dynamic interactive context, is the consistency of both quantitative and qualitative aspects (which are distinct) in terms of each other.
This thereby entails that the linear notion of consistency, relating to the primes - that is assumed to hold in analytic quantitative terms as individual independent "building blocks" with respect to the real numbers on this line - must be balanced by a complementary linear notion of consistency that is assumed to hold in a holistic qualitative manner with respect to these same numbers (through their interdependent relationship with each other). And as we have seen this complementary notion, relating to the holistic qualitative nature of the number system, is indirectly expressed - in an analytic quantitative manner - through the mathematical notion of the imaginary.
Therefore, this now entails that the (direct) quantitative notion of consistency with respect to numbers on the real number line intimately depends on the corresponding (indirect) quantitative notion of consistency with respect to the zeros on the imaginary number line i.e. that all such zeros should lie on the imaginary line.
And because of the requirement that objective reality in external terms be - ultimately - exactly matched by subjective interpretation in internal terms (which we dealt with in the previous entry) this poses the additional requirement that this imaginary line be drawn through 1/2 (on the real axis).
In other words, the fundamental significance of the Riemann Hypothesis is that its truth is a necessary requirement for the presumption of quantitative consistency with respect to the real number system.
In other words, from a proper dynamic interactive perspective, we are not entitled to presume quantitative consistency without reference to the complementary qualitative aspect of number.
Equally - and very importantly - we are not entitled to presume qualitative consistency without reference to the complementary quantitative aspect of number.
So the truth of the proposition that all the zeros line on the imaginary line (through 1/2) depends on the corresponding truth that all real numbers likewise lie on a line.
And the proposition that all real numbers lie on this line equally depends on the truth that all the zeros lie on the imaginary line (through 1/2).
And there is a necessary uncertainty principle in operation here.
In order to approach the quantitative extreme of rational absolute type understanding of mathematical relationships, the qualitative aspect must be rendered so "fuzzy" as to be no longer even identifiable. And this is the position that characterises conventional mathematical understanding, where its distinctive qualitative aspect is no longer even recognised (in formal terms).
And likewise to approach the opposite extreme of a purely relative type understanding of mathematical relationships, the quantitative aspect in turn must be rendered so "fuzzy" as to be no longer even identifiable. And this represents purely intuitive understanding in a psycho spiritual manner, where the ultimate relationship as between the primes and natural numbers is now directly "seen" in such a synchronous manner (that neither can maintain a distinct identity).
So all mathematical activity must lie between these two extremes, where purely absolute and purely relative understanding are respectively approached, implying always both quantitative notions (of relative independence) and qualitative notions (of relative interdependence) respectively.
Because both of these possible extremes (analytic and holistic aspects) are two sides of the same coin and - ultimately inextricably interdependent with each other - we cannot hope to prove either proposition (in a conventional analytic manner). So the Riemann Hypothesis is not capable of proof (or disproof) though in itself this is much less importance than its (unrecognised) significance.
In other words, the true fundamental issue underlying all Mathematics, relates to the prior consistency as between its quantitative (analytic) and qualitative (holistic) aspects.
And this is the issue to which the Riemann Hypothesis - when properly understood - directly points.
Therefore, we must assume the truth of the Riemann Hypothesis to - literally - maintain our faith in the subsequent consistency of all quantitative relationships. Equally - though not properly appreciated - we must assume the truth of the number line (i.e. that all real numbers lie on this line) to maintain faith in the subsequent consistency of all qualitative relationships.
And because of their distinctive nature, neither of these aspects can be proved in terms of each other.
So underlying the interpretation of all mathematical relationships is - implicitly - an initial massive act of faith in the subsequent consistency (in both quantitative and qualitative terms) of the whole enterprise.