Study and Consequences of the Ⓢ Function on the Riemann Hypothesis
Mohamed Sghiar
Mohamed Sghiar, 9 allée Capitaine Jean Bernard Bossu, Talant (Bourgogne), France.
Manuscript received on 22 December 2025 | First Revised Manuscript received on 31 December 2025 | Second Revised Manuscript received on 18 March 2026 | Manuscript Accepted on 15 April 2026 | Manuscript published on 30 April 2026 | PP: 5-8 | Volume-6 Issue-1, April 2026 | Retrieval Number: 100.1/ijam.A123106010426 | DOI: 10.54105/ijam.A1231.06010426
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Abstract: I will study the Sghiar’s function Ⓢ:(X,z)⟼∏_(p∈P)1/(1-X/p^z ), P the set of prime numbers. Which is an extension of the Riemann zeta function. The classical form of the Riemann zeta function and its Euler product are well known in analytic number theory, and I will show that : ζ(s)=0 and Re(s)>1/2⇒Ⓢ=0. We deduce the proof of the Riemann Hypothesis. MSC code: 11M26 ; 97F60 ; 32A10
Keywords: Prime Number, Holomorphic Function, the Riemann Hypothesis.
Scope of the Article: Number Theory