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

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 is an element of (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.