Sunday, October 11, 2015

Thoughts on the Riemann Hypothesis

The Riemann Hypothesis is a mathematical conjecture that the (nontrivial) zeros of the Riemann zeta function all have real parts of 1/2 on the complex plane. Its proof (or disproof) is a millennial prize problem worth a fair amount of money and everlasting fame. Thus it has become over the 150+ years since its inception by Bernhard Riemann, a landmark for mathematic pursuit, but despite the efforts of the greatest minds in mathematics, the hypothesis remains a conjecture.

So I offer a naïve suggestion.

In the work, Principia Mathematica, Whitehead and Russell spend several pages developing the notion of cardinal and ordinal couples to the conclusion that 1 + 1 = 2. That is, there is a dimension of numbers with order arising out of the process of addition together with the concept of unity. Multiplication is another dimension related inextricably to that of addition as the replication of the now existent numbers by each other along the dimension of order. The primes are simply byproducts of this relation.

The Riemann Hypothesis seems complex, but it is a result of the structure of multiplication over addition and nothing more. Two dimensions are required to solve polynomials--the expression of addition and multiplication together--and so the complex plane is two-dimensional (the imaginary number, i,  is just a symbol that marks the relation). No more and no less. The fact that multiplication is an operation quasi-independent of addition necessitates two dimensions for the expression of the solution of a polynomial expression once we have allowed that numbers have a structure that is ordinal.

My guess is that we will eventually find that the density of the primes is nothing more than a representation of the structure of addition and multiplication expressed in spatial dimensions similarly to how we view fractals--apparently complex, but exactly only the generating formula and nothing more in its essence.

The Riemann Hypothesis could not exist without the Euler product formula which is itself an expression of the Sieve of Eratosthenes which is nothing more than an expression of the artifacts of multiplication over the generated dimension of addition.

In short, the Riemann Hypothesis, is nothing more and nothing less than a statement that there is a midpoint created by the new ordinal relation of 1 + 1 and it occurs at 1/2 the distance between the unity of multiplication and the unity of addition. It cannot be anything different and it cannot be anything more or less.

Of course, this is just a thought.