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

This article 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 $\sigma \in (1\mathrm{/}2,1)$ 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 $\sigma >1\mathrm{/}2$ and we show that there is a connection with the zeros of ${\zeta }^{\mathrm{\prime }}(\sigma +it)$. Moreover, we clarify under which conditions the real and the imaginary part of the zeta-function are jointly universal.