Numerical Testing of the Riemann Hypothesis

Abstract: 

A sequence of remarkably successful calculations has shown that the first 100,000,000,000 zeros of the zeta function zeta(s) in the upper half of the strip 0<Re(s)<1 have real part one half.  This note outlines a quite independent method of testing the Riemann Hypothesis (RH).  Andre Weil's quadratic functional (1953) on a suitable space of functions on the group of positive real numbers can be evaluated for what have to pass for step functions and the positive definiteness of some of a family of symmetric matrices determined.  If any of these turned out not to be positive definite the RH would be disproved.  No such example was found!  Conversely, if all of these are shown to be positive definite the RH would have been verified.