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 $t\mapsto \zeta (\frac{1}{2}+it)$. Moreover, we show that the curve $t\mapsto (\zeta (\frac{1}{2}+it),{\zeta }^{\prime }(\frac{1}{2}+it))$ is not dense in $C^2$.

15 pages

References in corpus (3)

• More than 41% of the zeros of the zeta function are on the critical line
• Random matrix theory and discrete moments of the Riemann zeta function
• On simple zeros of the Riemann zeta-function

Cited by in corpus (4)

• Value-Distribution of the Riemann Zeta-Function along its Julia Lines
• On the Vertical Distribution of Values of $L$-functions in the Selberg Class
• Spirals of Riemann's Zeta-Function --Curvature, Denseness, and Universality--
• The $a$-points of the Riemann zeta-function and the functional equation