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Sunday, March 31, 2013

On The Riemann Zeta Function

The Riemann  function depends on the complex variable , and also in a deep way, on the nature of natural numbers. Every natural number can be written as a product of primes, but no prime can be written as a product of more than one natural. This deep asymmetry is at the bottom of Riemann's hypothesis.

The zeroes of this function, away from the real axis, have the property of a unique real part: 


Primes play the role of elementary particles in Physics. All matter (except dark), is composed of quarks and leptons, but as far as we know, these particles are not composite.

Primes then, seem like natural mental constructs, for a mathematical theory of Physics.

Neither Riemann's Conjecture, nor this Physical theory are understood today.

One connection with Physics, is the spectral decomposition of an operator. Operator Zeta Function. One can see the nature of the problem this way: There cannot be a formula which gives us all prime numbers. We will always be looking for the next prime number. This may be related to mathematical logic problems, like the no-halting problem.

Read Schumayer & Hutchinson, and Menezes, Svaiter, & Svaiter.

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