Yochay Jerby, Holon Institute of Technology, Israel

Abstract

The Riemann hypothesis is a question regarding the solutions of the transcendental equation ζ(s)=0, that is the zeros of the Riemann zeta function. The starting point of our talk is a remarkable connection between the zeros of zeta and the solutions of the much simpler equation χ(s)+1=0, whose non-trivial solutions could be completely described in closed form and all lie on the critical line. We will explain that the two equations are related by a sequence of functions ζN(s) called $N$-th sections of zeta. For N=1 one has ζ1(s)=1/2(1+χ(s)) while for N=[Im(s)/2] the section ζN(s) is approximately ζ(s), up to a small error. Studying the dynamical change of the zeros of ζN(s) with respect to N we will show that zeros can go off the critical line only if a process of collision occurs between two consecutive zeros at a certain stage. We will also suggest a method of re-arranging the dynamics so that collisions could be avoided altogether, that is, in a way expected to keep the zeros on the critical line for any N, including the final stage where they essentially coincide with the zeros of zeta. If time permits we will also discuss how the suggested viewpoint relates to various classical topics such as: Gram's law, the Davenport-Heilbronn function and the Montgomery pair correlation conjecture.

Link to Video: https://vimeo.com/631211084

Presented Via Zoom: https://rutgers.zoom.us/j/94346444480

Password: 6564120420

For further information see: https://sites.math.rutgers.edu/~zeilberg/expmath/