Study and Consequences of the Ⓢ Function on the Riemann Hypothesis
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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
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J. B. Conrey, The Riemann Hypothesis, Notices of the AMS, 2018, DOI: http://doi.org/10.1090/noti1667
K. Soundararajan, Moments of the Riemann zeta function, Annals of Mathematics, 2017, DOI: http://doi.org/10.4007/annals.2017.185.2.3
A. Harper, Bounds on the Riemann zeta function, Proceedings of the London Mathematical Society, 2019, DOI: http://doi.org/10.1112/plms.12225
M. Radziwiłł and K. Soundararajan, Moments and distribution of central L-values, Inventiones Mathematicae, 2020,
DOI: http://doi.org/10.1007/s00222-020-00950-6
P. Sarnak, Three lectures on the Möbius function randomness, 2021, Available at: https://publications.ias.edu/sarnak/paper/563
E. M. Stein and R. Shakarchi, Complex Analysis, Princeton University Press. Published : Apr 27, 2003
.https://press.princeton.edu/books/hardcover/9780691113852/complexanalysis