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

TL;DR

The paper proves the existence of infinitely many simple roots of the equation ζ(s)=a with large imaginary parts, offers a heuristic on the regularity of the ζ curve, and shows the non-density of a related curve in C^2.

Contribution

It establishes the infinitude of simple roots for ζ(s)=a with large imaginary parts and analyzes the geometric properties of ζ on the critical line.

Findings

Infinitely many simple roots of ζ(s)=a with arbitrarily large imaginary parts.

Heuristic interpretation of the regularity of the ζ curve.

The curve (ζ(1/2+it), ζ'(1/2+it)) is not dense in C^2.

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 $C^2$.