The Riemann hypothesis illuminated by the Newton flow of ζ☆We dedicate this paper to the memory of Richard Lewis Arnowitt and his many contributions to general relativity and high energy physics.

Abstract

Abstract We analyze the Newton flow of the Riemann zeta function ζ and rederive in an elementary way the Riemann–von Mangoldt estimate of the number of non-trivial zeros below a given imaginary part. The representation of the flow on the Riemann sphere highlights the importance of the North pole as the starting and turning point of the separatrices, that is of the continental divides of the Newton flow. We argue that the resulting patterns may lead to deeper insight into the Riemann hypothesis. For this purpose we also compare and contrast the Newton flow of ζ with that of a function which in many ways is similar to ζ, but violates the Riemann hypothesis.