All Complex Zeros of the Riemann Zeta Function Are on the Critical Line: Two Proofs of the Riemann Hypothesis

TL;DR

This paper provides two independent proofs confirming the Riemann Hypothesis by demonstrating all non-trivial zeros of the Riemann Zeta Function lie on the critical line, using classical complex analysis theorems.

Contribution

It introduces two novel proofs of the Riemann Hypothesis that rely solely on well-known theorems in complex analysis, avoiding the functional equation.

Findings

All non-trivial zeros are on the critical line

The admissible domain of zeros is the critical line

Proofs are accessible and do not depend on the functional equation

Abstract

I present two independent proofs of the Riemann Hypothesis considered by many the greatest unsolved problem in mathematics. I find that the admissible domain of complex zeros of the Riemann Zeta Function is the critical line. The methods and results of this paper are based on well-known theorems on the number of zeros for complex value functions (Jensen, Titchmarsh, Rouche theorems), with the Riemann Mapping Theorem acting as a bridge between the Unit Disk on the complex plane and the critical strip. By primarily relying on well-known theorems of complex analysis our approach makes this paper accessible to a relatively wide audience permitting a fast check of its validity. Both proofs do not use any functional equation of the Riemann Zeta Function, except leveraging its implied symmetry for non-trivial zeros on the critical strip.