A Possible Proof Of The Riemann Hypothesis
Year: 2024
Type: article
Abstract: In this paper, I present my approach to the Riemann Hypothesis, which potentially provides a proof of it. I start by simplifying the equality Zeta(s)=0 by expressing |Zeta(s)|^2=0 as Re(Zeta(s))^2 + Im(Zeta(s))^2 = 0, expressed with cosine and sine functions, then I use trigonometric identities to further break down the sum into a sum of squares purely relying on Re(s) and a double sum relying on both Re(s) and Im(s). Using the bounded nature of cosine, I reformulate the equality as problem of quadratic equations, to get rid of the dependence on Im(s), I then consider Re(s) as a map a[n] (n being the number of terms in Zeta(s)), rather than a fixed value (since we deal with infinity), It has already been proved that for all the nontrivial zeros, their real part is in ]0,1[, So, I then use integrals and asymptotic analysis to observe three cases of asymptotic behaviours: when a[n] converges to a real number in ]0,1/2[ when a[n] converges to a real number in ]1/2,1[ when a[n] converges to 1/2 (where we distinguish some more cases) Then I conclude that the only logically consistent case is when a[n] converges to 1/2.
Author: KAPLAN, Yunus-Emre
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W6893076717
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