arXiv subjects

Jon Breslaw

Publications and source records attributed to Jon Breslaw.

2 recordsLinked to original sources

arXiv2015

A Reformulation of the Xi Function

This paper proposes a reformulation of the Riemann Xi function in order to investigate its properties. The reformulated function, which depicts the Xi function as the weighted sum of incomplete gamma functions, is validated, and a number of properties are established. These properties are then used to analyze the intersection of the real and imaginary zero level curves. It is shown that a pair of zeros off the critical line is not consistent with these properties, thus validating the Riemann hypothesis.

math.GM↗

arXiv2009

Towards a statistical proof of the Riemann Hypothesis

Using the $ζ$ functional equation and the Hadamard product, an analytical expression for the sum of the reciprocal of the $ζ$ zeros is established. We then demonstrate that on the critical line, $|ζ|$ is convex, and that in the region $0 < \Re(s) \leq 0.5$, $|ζ|$ has a negative slope, given the RH. In each case, analytical formulae are established, and numerical examples are presented to validate these formulae.

math.GM↗