An electrostatic depiction of the validity of the Riemann Hypothesis and a formula for the N-th zero at large N

TL;DR

This paper models the Riemann zeta function using an electrostatic analogy, deriving a formula for high zeros on the critical line and providing insights into the Riemann Hypothesis through a vector field approach.

Contribution

It introduces a novel electrostatic framework to analyze the Riemann zeta function and derives a new formula for the n-th zero at large n, improving zero estimation accuracy.

Findings

Derived a formula for the n-th zero on the critical line for large n.

Showed that off-line zeros cause frustration in the electrostatic model.

Estimated the 10^{10^6}-th zero with high accuracy.

Abstract

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). E satisfies the conditions for a static electric field. The structure of E 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 + or - 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 this "electric" field. We formulate our argument more precisely in terms of the potential Phi satisfying E = - gradient Phi, and construct Phi explicitly. One outcome of our analysis is a formula for the n-th zero on the critical line for large n expressed as the solution of a simple transcendental equation. Riemann's counting formula for the number of zeros on the entire critical strip can be derived from this formula. Our result is much stronger than Riemann's counting formula, since it provides an estimate of the n-th zero along the critical line. This provides a simple way to estimate very high zeros to very good accuracy, and we estimate the 10^{10^6}-th one.