On the roots of the equation $\zeta(s)=a$

Abstract

Given any complex number $a$, we prove that there are infinitely many simple roots of the equation $\zeta(s)=a$ with arbitrarily large imaginary part. Besides, we give a heuristic interpretation of a certain regularity of the graph of the curve $t\mapsto \zeta({1\over 2}+it)$. Moreover, we show that the curve $t\mapsto (\zeta({1\over 2}+it),\zeta'({1\over 2}+it))$ is not dense in $C^2$.