The Extension of the Riemann’s Zeta Function

In mathematics, the search for exact formulas giving all the prime numbers, certain families of prime numbers or the n-th prime number has generally proved to be vain, which has led to contenting oneself with approximate formulas 8. The purpose of this article is to give a new proof of the Riemann hypothesis 4-which is closely related to the distribution of prime numbers- by y introducing S^ a new extension of the of the Riemann zeta function.