Proof that the real part of all non-trivial zeros of Riemann zeta function is 1/2

TL;DR

This paper claims to prove the Riemann hypothesis by analyzing the integral form of the completed zeta function, demonstrating that all non-trivial zeros lie on the critical line with real part 1/2.

Contribution

It provides a detailed proof of the Riemann hypothesis by contour deformation and singularity analysis of the completed zeta function.

Findings

All non-trivial zeros have real part 1/2.

Singularities are canceled except at the zeros on the critical line.

The proof relies on integral contour deformation and residue analysis.

Abstract

This article proves the Riemann hypothesis, which states that all non-trivial zeros of the zeta function have a real part equal to 1/2. We inspect in detail the integral form of the (symmetrized) completed zeta function, which is a product between the zeta and gamma functions. It is known that two integral lines, expressing the completed zeta function, rotated from the real axis in the opposite directions, can be shifted without affecting the completed zeta function owing to the residue theorem. The completed zeta function is regular in the region of the complex plane under consideration. For convenience in the subsequent singularity analysis of the above integral, we first deform and shift the integral contours. We then investigate the singularities of the composite elements (caused by polynomial integrals in opposite directions), which appear only in the case for which the distance between the contours and the origin of the coordinates approaches zero. The real part of the zeros of the zeta function is determined to be 1/2 along a symmetry line from the singularity removal condition. (In the other points, the singularities are adequately cancelled as a whole to lead to a finite value.)