Spirals of Riemann’s Zeta-Function — Curvature, Denseness and Universality

Abstract

Abstract This paper deals with applications of Voronin’s universality theorem for the Riemann zeta-function $\zeta$ . Among other results we prove that every plane smooth curve appears up to a small error in the curve generated by the values $\zeta(\sigma+it)$ for real t where $\sigma\in(1/2,1)$ is fixed. In this sense, the values of the zeta-function on any such vertical line provides an atlas for plane curves. In the same framework, we study the curvature of curves generated from $\zeta(\sigma+it)$ when $\sigma>1/2$ and we show that there is a connection with the zeros of $\zeta'(\sigma+it)$ . Moreover, we clarify under which conditions the real and the imaginary part of the zeta-function are jointly universal.