University of Sydney Algebra Seminar

Peter Donovan (University of New South Wales)

Friday 22nd March, 12:05-12:55pm, Carslaw 373

Numerical testing of the Riemann Hypothesis

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 \(0 < \mathfrak{R}(s) < 1\) have real part \(\frac{1}{2}\). This talk outlines a quite independent method of testing the Riemann Hypothesis (RH). André 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 deniteness of some of a family of symmetric matrices determined. If any of these turned out not to be positive denite the RH would be disproved. No such example was found! Conversely, if all of these are shown to be positive denite the RH would have been verifed.