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 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