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 elements for a proof of it. I start by simplifying the equality Eta(s)=0 (which is implied by the nullness of the continued Zeta(s)) by expressing |Eta(s)|^2=0 as Re(Eta(s))^2 + Im(Eta(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 the cosine function, I reformulate the equality as a problem of quadratic equations, to get rid of the dependence on Im(s) as well as the oscillations it induces, I then consider Re(s) as a map a[n] = Re(s[n]) (n being the number of terms in Zeta(s) or Eta(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: I MORE OR LESS FINISHED THE CHANGES 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.Or maybe not, but good attempt, I guess. IMPORTANT NOTE: I rushed to complete the last part with the corrections terms, I'll get into better detail when I have time. I did some typo corrections in this version, I'll keep making things more explicit. For this version of the document I'll use the correction terms only in the comparison parts and not in the integral expressions, I'll do it in the coming versions. I'm really sorry for the way it's obvious the last part is rushed and should be presented better, it's just because a lack of time. I'll update it whenever possible Edit: There are little mistakes, I keep correcting them, I'm sorry, the reasoning is there though. Now I am making things more clearly expressed
Author: KAPLAN, Yunus-Emre
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W6893233138
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