This is dedicated to the good people at Facebook, YouTube, and Google. Abstract We indirectly prove the Riemann Hypothesis by proving the Nicholas equivalency.
The Riemann Hypothesis states that the nontrivial zeros of the Riemann Zeta Function have a real part of^12. We examine the following equivalency [1] established by Nicholas [2]:
φN(Nkk)> eγloglogNk
where
- γis the Euler Mascheroni constant,
- φis the Euler totient function, and
- Nkis the primorial of the firstkprimes The Nicholas equivalency states that if this relationship holds for allk≥1 then the Riemann Hypothesis is true.
We take the M ̈obius inversion [7] ofφ(Nk) on the left and we have:
N Nk
kΣd|nμ(dd)
which reduces to:
1
(1−^11
p 1 )(1−p^12 )··(1−p^1 k)> e
γloglogNk
We apply Merten’s Theorem [3] [4]:
limn→∞lnp^1 n∏∞k=1(1−^1 p (^1) k)=eγ To get: log(pk)eγ=eγloglogNk On the right side, by the Chebyshev function [6] the log of the primorial [5] is: ln(n#) =θ(n) which asymptotically approachesnfor large values ofn[8], giving us: log(pk)eγ> eγlog(k) Simplifying, we have: loglog((pkk))> 1 Which is true for all values ofk, proving the Riemann Hypothesis.
[1] Conrey, J. Brian and Farmer, David W.,Equivalences To The Riemann Hypothesis, From Way- back Machine, https://web.archive.org/web/20120731034246/http://aimath.org/pl/rhequivalences [2] Jean-Louis Nicolas,Petites valeurs de la fonction dEuler.J. Number Theory 17 (1983), no. 3, 375388 [3] Sondow, Jonathan and Weisstein, Eric W.Mertens Theorem. From MathWorld–A Wolfram Web Resource. http://mathworld.wolfram.com/MertensTheorem.html [4] Mertens, F.Ein Beitrag zur analytischen Zahlentheorie.J. reine angew. Math. 78, 46-62, 1874. [5] Weisstein, Eric W. Prime Products. From MathWorld–A Wolfram Web Resource. http://mathworld.wolfram.com/PrimeProducts.html [6] Weisstein, Eric W. Chebyshev Functions. From MathWorld–A Wolfram Web Resource. http://mathworld.wolfram.com/ChebyshevFunctions.html [7] Lynn, Ben. Mbius Inversion. Stanford Applied Cryptography Group Web Page. https://crypto.stanford.edu/pbc/notes/numbertheory/mobius.html [8] Anonymous.Primorial.Wikipedia. https://en.wikipedia.org/wiki/Primorial
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