A focus on the Riemann's Hypothesis

TL;DR

This paper explores the properties of the Riemann zeta function, distinguishes it from its analytical extension, and demonstrates the existence and uniqueness of zeros with real part 1/2, contributing to understanding the Riemann Hypothesis.

Contribution

It shows that the Riemann zeta function and its analytical extension are distinct and proves the existence and uniqueness of zeros with real part 1/2.

Findings

Riemann zeta function and its extension are distinct

Existence of zeros with real part 1/2 is established

Uniqueness of zeros with real part 1/2 is demonstrated

Abstract

Riemann's hypothesis, formulated in 1859, concerns the location of the zeros of Riemann's Zeta function. The history of the Riemann hypothesis is well known. In 1859, the German mathematician B. Riemann presented a paper to the Berlin Academy of Mathematic. In that paper, he proposed that this function, called Riemann-zeta function takes values 0 on the complex plane when s=0.5+it. This hypothesis has great significance for the world of mathematics and physics. This solutions would lead to innumerable completions of theorems that rely upon its truth. Over a billion zeros of the function have been calculated by computers and shown that all are on this line s = 0.5+it. In this paper, we initially show that Riemann's (Z\^eta) function and the analytical extension of this function called (Aleph)) are distinct. After extending this function in the complex plane except the point s=1, we will show the existence and then the uniqueness of real part zeros equal to 1/2.