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 I corrected some careless mistake and added a remark about an alternative and more straightforward way of figuring out the asymptotic expansions, underlying that the outcome would be exactly the same anyway. I'm not far from the final version (I hope).