And once again, because in the very dynamics of understanding these two aspects (quantitative and qualitative) are indissolubly linked, the Riemann Hypothesis - when correctly interpreted - necessarily relates to both aspects.
So, when appropriately interpreted in holistic mathematical terms, the Riemann Zeta Function (and Hypothesis) has a remarkable relevance for the precise qualitative understanding of certain psycho-spiritual dynamics at the most advanced levels of human development (which complements corresponding understanding of quantum physical interactions at the sub-atomic levels of matter).
One of the problems with explaining the Riemann Hypothesis is that its fascination comes from its deep connection to prime numbers, but its definition is in terms of complex analysis which requires a fair deal of undergraduate mathematics to understand -- and that is before you even got started to grasp what the heck the zeta-zeros have to do with the distribution of primes. My "cocktail party explanation" of the Riemann Hypothesis would usually be something like: "The prime numbers are as equally distributed as you could wish for." But there is actually a surprisingly easy interpretation of the Riemann Hypothesis: "Prime numbers behave like a random coin toss."
So a major philosophical dilemma - which is central to difficulties in unravelling the Riemann Hypothesis - is that Conventional Mathematics still attempts to interpret complex numbers through a qualitative approach that is solely real.
The significance of this is that the (with which the Riemann Hypothesis is intimately linked) employs complex dimensions and generates many results that have no strict meaning in quantitative terms.
The problem is, the gap occurs right at the beginning, and so it's hard to fill that gap because you don't see what's on the other side of it." It is my firm belief that the failure to solve the Riemann Hypothesis relates to a deep philosophical problem that strikes at the very heart of what conventionally is understood as mathematics.
It's a gaping hole in our understanding..." Finally Hugh Montgomery :
"Sometimes I think that we essentially have a complete proof of the Riemann Hypothesis except for a gap.
Perhaps the most famous earlier school associated with the development of number theory is that that of the Pythagoreans and at least two of their contributions - when appropriately interpreted - can be shown to have an intimate bearing on the Riemann Hypothesis.
And this lack of appropriate philosophical understanding, as I will attempt to demonstrate, is central to the failure to properly appreciate what is implied by the Riemann Hypothesis.
In fact as we shall see later, the Riemann Hypothesis is essentially a statement regarding the basic mathematical relationship of quantitative to qualitative interpretation.
I recently spent some timeÂ on the formidable website which explains mathematical ideas, some important, some recreational, in short and accessible videos. Definitely worth checking out. One of the videos that is most relevant to us explains the Riemann Hypothesis:
Indeed this realisation of the intimate connection, as between prime numbers and the behaviour of quantum particles, has now come to be recognised in a dramatic fashion through the search for a solution to the Riemann Hypothesis.
The (with which the Riemann Hypothesis is intimately related) starts in a very simple series that again is famously associated with the Pythagoreans.
And when we properly interpret such results in a qualitative manner (which is the way they are designed to be decoded) the true nature of prime number behaviour is again revealed as entailing the interaction of two distinct logical systems, linear and circular, which in turn paves the way for ready resolution of the Riemann Hypothesis.
For example Brian Conrey  :
"The Riemann Hypothesis is the most basic connection between addition and multiplication that there is, so I think of it in the simplest terms as something really basic that we don't understand about the link between addition and multiplication."And Alain Connes  in somewhat similar fashion:
"The Riemann Hypothesis is probably the most basic problem in mathematics, in the sense that it is the intertwining of addition and multiplication.