A Possible Proof Of The Riemann Hypothesis - Yunus-Emre KAPLAN.pdf

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] of complex numbers tending to a real number (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

Then I conclude that the only logically consistent case is when a[n] converges to 1/2.

Actually, I even find that the real part has to be 1/2 all the way, and that it is actually the imaginary part that has a rate of convergence.

NOTES:

In the current version, I basically made a "clean up for accuracy".

In the previous versions on Zenodo:
(link for the current version: https://zenodo.org/records/14722273)

- I had completed my reasoning regarding some conditions I rely on

- I had also made changes on the way I handle Taylor expansions, and deleted redundant parts.

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

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

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.