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 ζ (Zêta) function and the analytical extension of this function called ℵ (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.
Key words: Riemann’ hypothesis, Hadamard product, zeta function
Copyright © 2020 Author(s) retain the copyright of this article.
This article is published under the terms of the Creative Commons Attribution License 4.0