The holomorphic flow of Riemann's function ξ(z)

Abstract

The holomorphic flow of Riemann's xi function is considered. Phase portraits are plotted and the following results, suggested by the portraits, proved: all separatrices tend to the positive and/or negative real axes. There are an infinite number of crossing separatrices. In the region between each pair of crossing separatrices—a band—there is at most one zero on the critical line. All zeros on the critical line are centres or have all elliptic sectors. The flows for ξ(z) and cosh(z) are linked with a differential equation. Simple zeros on the critical line and Gram points never coincide. The Riemann hypothesis is equivalent to all zeros being centres or multiple together with the non-existence of separatrices which enter and leave a band in the same half plane.