The Extension of the Riemann’s Zeta Function
Mohamed Sghiar

Mohamed Sghiar, Department of Mathématiques, Faculté des Sciences Mirande, Université de Bourgogne Dijon, France.

Manuscript received on 04 August 2023 | Revised Manuscript received on 08 August 2023 | Manuscript Accepted on 15 March 2024 | Manuscript published on 30 May 2024 | PP: 4-7 | Volume-10 Issue-7, March 2024 | Retrieval Number: 100.1/ijbsac.A05020910223 | DOI: 10.35940/ijbsac.A0502.10070324

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Abstract: 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.

Keywords: Prime Number, number theory, distribution of prime numbers, the law of prime numbers, the Gamma function, the Mertens function, quantum mechanics, black Holes, holomorphic function, Hilbert-Polya’s conjecture, the Riemann hypothesis.
Scope of the Article: Functional Analysis