1305.2613 (Andre LeClair)

An electrostatic depiction of the validity of the Riemann Hypothesis    [PDF]

Andre LeClair

We construct a vector field E from the real and imaginary parts of an entire function xi (z) which arises in the quantum statistical mechanics of relativistic gases when the spatial dimension d is analytically continued into the complex z plane. This function is built from the Gamma and Riemann zeta functions and is known to satisfy the functional identity xi (z) = xi (1-z). We describe how this non-trivial identity can be demonstrated using quantum field theory arguments in a cylindrical geometry, where it relates finite temperature black body physics to the Casimir energy on a circle. The vector field E satisfies the conditions for a static electric field. The structure of this "electric field" in the critical strip is determined by its behavior near the Riemann zeros on the critical line Re (z) = 1/2, where each zero can be assigned a plus or minus vorticity of a related pseudo-magnetic field. Using these properties, we show that a hypothetical Riemann zero that is off the critical line leads to a frustration of the electric field, which is to say, an incompatibility with the electric field pattern that is a consequence of the infinite number of zeros along the critical line. We also reformulate our argument in terms of the potential phi satisfying E = - gradient of phi and construct phi explicitly.

View original: http://arxiv.org/abs/1305.2613