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] 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 is a converging map.

NOTES:

I made big corrections in this version, in the previous versions (last version on zenodo: https://zenodo.org/records/14642065) even though I had mentioned the alternative method of approximating with integrals first and expanding last, I used expanding first-integrate last instead; it actually works fine for lim a[n] in ]0,1/2] but that raises a problem for lim a[n] > 1/2.

So I used the integrate first-expand last method for both lim a[n] != 1/2 cases, and the expand first-integrate last for the lim a[n] = 1/2 case only, to clarify that the lambda in this precise case is the Euler-Mascheroni constant noted as a gamma 𝛾∼0.5772156649.

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.