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.

Before starting I define s[n]=a[n]+ib, a zero to the truncated Eta function.

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 a[n] and a double sum relying on both a[n] and b.

The emergence of the sum of squares sum(1/k^(2*a[n]),k=1..n) is what shows us we're on the right path, for we can compare it with (n^(1-2*a[n])-1)/(1-2*a[n]) if lim a[n] !=1/2 or ln(n) if lim a[n]=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 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.

And this imaginary rate of convergence in that pseudo-real-part, is strange if we take into account that we deemed a[n] as a real part when we used De Moivre's formulas.

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. If anyone has insights into this mystery, I'd greatly appreciate an explanation.

The clumsy amateur mathematician that I am is a little puzzled now.

- In this version, I corrected the way I expand the sums in the 1/2 case, and I started considering the real part as a sequence right from the beginning instead of waiting for part 3.

- In recent previous versions I had improved accuracy, and 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.