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: 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. NOTE: I made corrections again, I re-included the part where I deal with infinite orders of expansion, it actually works very well, and there's a wonderful plot twist: It is actually the imaginary part Im(s) that is a map with a convergence rate, the real part is plainly Re(s)=1/2. I keep making corrections (and thus new versions) as I spot errors, ambiguous expressions and incomplete statements. Above all now, I have to modify many statements about the real part being a map, now that my conclusions proved it was actually constant all the way, but just the imaginary part that was a map. In this version, as promised in the previous version, I removed the parts where I deal with finite orders of approximations because it was somehow redundant, but only for the ]0,1/2[ and ]1/2,1[ cases; I kept it for the 1/2 case because I deem it an essential transition to understand what happens as p->infinity I'm not far from the final version (I hope).