A Possible Proof Of The Riemann Hypothesis
Creators: name: KAPLAN, Yunus-Emre
Publisher: name: Zenodo
Year: 2025
Subjects: Riemann Hypothesis
Asymptotic Analysis
Prime Numbers
Imbricated Quadratic Equations
Integrals
Zeta Function
Eta Function
Taylor Expansion

description: EDIT: I corrected my paper but couldn't make a new version on January the 18th, 2025, there likely is a bug on the platform, so here is a figshare link of my new version:

https://figshare.com/articles/preprint/A_Possible_Proof_Of_The_Riemann_Hypothesis_-_Yunus-Emre_KAPLAN_pdf/28235450?file=51780395

DOI:https://doi.org/10.6084/m9.figshare.28235450.v1

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This study is my approach to the Riemann Hypothesis, which potentially provides elements for a proof of it, or even an actual proof.

I start by breaking down 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:



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.

NOTES: 

I made corrections again, I re-included the part where I deal with infinite orders of expansion, it actually works very well, and there's a wonderful plot twist:

It is actually the imaginary part Im(s) that is a map with a convergence rate, the real part is plainly Re(s)=1/2.

I keep making corrections (and thus new versions) as I spot errors, ambiguous expressions and incomplete statements.

In this version, I corrected a statement about the building of epsilon_n in the Taylor expansion of order p in the last a_n->1/2 part, and added at the end of the paper that the lambda in the 1/2 case is the Euler-Mascheroni constant gamma 𝛾∼0.5772156649.

I'm not far from the final version (I hope).

I'm currently correcting the way I handle the Taylor expansions, that changes nothing for the rest of the reasoning. Expanding the sum first and approximating with integrals raises a problem for a_n converging to a real ]1/2,1[. So for the cases where a_n doesn't converge to 1/2, in the next version, I will directly approximate the sum involving a rate of convergence epsilon_n with integrals involving epsilon_n as well.

Feedback would be appreciated at: emrekaplan1978@gmail.com

Despite the quite consequent number of downloads, I've received no feedback so far, positive or negative, despite my email address being public.So a review would really be most welcomed.