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. IMPORTANT NOTE: The behaviour of the sums in the 1/2 case, for an infinite order of expansion, behaves strangely, although there exist adequate rates of convergence for any fixed order of expansion. No order of expansion provides any solution when the real part is different from 1/2 (I should detail it in the upcoming version). This deserves some more investigation. In the upcoming versions, I may remove the parts where I was ignoring the correction terms, for it is uselessly redundant. Only the parts taking the correction terms in account shall be retained. I'm not far from the final version (I hope).