A Possible Proof Of The Riemann Hypothesis

In this paper, I present my approach to the Riemann Hypothesis, which potentially provides elements for a proof of it, or even an actual proof.

I start by simplifying 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:

                       Almost finished

• 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.

IMPORTANT NOTE: I keep making things more explicit, so don't be surprised if I keep making new versions.

I am currently deploying tremendous efforts to correct sentences and expressions and little mistakes here and there, and to remove ambiguities. My goal is to get as accurate as possible, I still have some changes to make, I'm not far from the final version. I corrected a careless mistake on a +/- sign.

As you can see, the amount of signs and calculations makes the careless mistakes really easy (I've corrected a good amount up to now).