Towards a resolution of the Riemann hypothesis

Abstract

This article contains work associated with a resolution of the Riemann hypothesis, following work by Taylor \cite{prt}, Lagarias and Suzuki \cite{lagandsuz} and Ki \cite{ki}, as well as Pustyl'nikov \cite{pust, pust2} and Keiper \cite{keiper}. Functions $\xi_+(s)$ and $\xi_-(s)$ are considered, for which it is known that all zeros lie on the critical line. The Riemann hypothesis itself pertains to the question of the location of the zeros of the sum and difference of $\xi_+(s)$ and $\xi_-(s)$, and this is investigated. An argument is developed which prima facie establishes the validity of the Riemann hypothesis. It adds to a necessary condition of Pustyl'nikov a sufficient condition. A second argument is discussed, which could have been accessible to Riemann.