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: 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).

The emergence of the sum of squares sum(1/k^(2*Re(s)),k=1..n) is what shows us we're on the right path, for we can compare it with (n^(1-2*Re(s))-1)/(1-2*Re(s)) if Re(s)!=1/2 or ln(n) if Re(s)=1/2, and since we want to prove Re(s)=1/2, that's a very good sign to see 1-2*Re(s) appear.

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 + o(1/ln(n)²), and that it is actually the imaginary part that has a slower (so more prominent) rate of convergence i*k*Pi(b+(ln(n)+4-lambda)/ln(n)²) where b=Im(s) once n has tended to infinity.

NOTES:

I'm currently in a big questioning phase. Only now do I remember why I was astonished that a[n] had an imaginary rate of convergence: because in the first part, when expressing |Eta(s)|^2=0 using the De Moivre's formula, I had handled it as a real.

- I actually rushed a little bit for the 1/2 case, the lambda constant isn't necessarily the Euler-Mascheroni constant gamma, I rushed as I expressed the sum of the correction terms as integrals. In the upcoming version, I will do like the cases where a!=1/2, I will integrate first, and then I shall expand to figure out an adequate epsilon[n]. That doesn't change the rest of the reasoning, in the end we just don't have the guarantee that lambda is gamma, and we have to replace gamma in the general expression of a nontrivial zero with a lambda belonging to [0,1].

- In this version, I changed some statements for accuracy purposes.

- In recent previous versions I had completed my reasoning regarding some conditions I rely on

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.