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

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 zeroes of the Riemann Zeta Function is the critical line. Methods and results of this paper are based on well-known theorems on the number of zeroes for complex value functions (Jensen, Titchmarsch, 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 zeroes on the critical strip.