Riemann’s Hypothesis New Proof

After some papers proving the RC (short for “Riemann’s Conjecture”, also known as the “Riemann’s Hypothesis”, RH), now the author provides a new proof, using the “Spira Criterion” that states “The RH is equivalent to the statement that if  >0.5 and t> 6.5 then |  (1-s)|> |  (s)|”. We use the concept of “transfer function” for control systems. This new proof is so simple that the author wonders why a great mathematician like Riemann did not see it; therefore F. Galetto thinks that somewhere in the purported proof there should be an error.

The other zeros are named nontrivial zeros: all the "known" zeros, computed up to now [up to 2004, 10 12 zeros have been computed, all on the Critical Line], are the complex numbers s=1/2 + it, with suitable values of t.
It satisfies the functional equation The properties of (s) are as follows: a) (s) has no zero for Re(s)>1; b) the only pole (s) is at s=1: it is simple and has residue 1; c) (s) has trivial zeros at the negative even integers z = -2, -4, -6, …. d) all the nontrivial zeros lie inside the region, named Critical Strip, 0  Re(s)  1 and are symmetric about both the vertical line, named Critical Line, Re(z)==1/2 and the real axis Im(z)=0: Riemann conjectured the so-called Riemann Hypothesis (RH), or Conjecture (RC): RC states All nontrivial zeros of (s) have real part  equal to 1/2 .
Many great mathematicians tackled this problem; we do not mention them, because they can be found in many books and papers.
If RC would be related to Physics, it would be considered a "universal law": up to 2004, 10 12 zeros have been computed, all on the Critical Line. zeros computed, all on the Critical Line (as to 2004) supports H 0 with that "high" CL.
But Mathematics asks much more than Statistics and Physics… Also a theorem of G. Hardy [Hardy's Theorem, 1914] that proved that "There are infinitely many zeros of (s) on the Critical Line" is not enough.
ALL the nontrivial zeros must be on the Critical Line, if one wants to prove RC.
The author, Fausto Galetto, is aware that (in these weeks) he has been affording a very important problem that great mathematicians have failed to prove.

The "Transfer Function" and the Spira Criterion
which can be written, in the form used in electronics, communication theory and control systems, where H(s) is the "Transfer Function" that links the Input (s) to the Output (1 − s).
The "Transfer Function" is the ratio Output/Input.
When =0 the "Transfer Function" depends only on the "frequency" ; from now on we use the symbol s=+j, as it is customary in electronics, communication theory and control systems: Squaring both side and taking the absolute value, we get This formula is the same as the following used in electronics, communication theory and control systems, where the function S(j) is the "Power Spectral Density". We name here |H(jω)| 2 the "Power Transfer Function".
In analogy to (3b) we define and name here |H(s)| 2 the "Generalised Power Transfer Function".

For n=1
www.scholink.org/ojs/index.php/asir  Figure 4 show the behaviour of |H(σ + jω)| for various values of (abscissa) σ and various ω (curves); all curves intersects at σ=0.5 and have ordinate |H(σ + jω)| = 1 We can confirm the previous result by writing it in a different way (Abramowitz et al., 1965 It is easily seen that (9) has a unique solution = 0.5; therefore the zeros of the Zeta Function (σ + jω) are all on the Critical Line = 0.5 + ; this is the same result found before by the Spira Criterion.
We can confirm the previous result by writing in a different way (Abramowitz et al., 1965)  It is easily seen that (9b) has a unique solution = 0.5; therefore …. (as before).

Conclusion
Titchmarsh proved that there are infinite zeros of the Riemann zeta function (s) in the Critical Strip, that is there are infinite values s k = k +i k such that (s k )=0, [0 <  k < 1]. G. Hardy [Hardy's Theorem, 1914] proved that "There are infinitely many zeros of (s) on the Critical Line": that fact was not conclusive, because ALL the zeros must be on the Critical Line.
In this paper, to prove RC we used first the "Spira Criterion" [with the Transfer Function H(s)] and second the "Generalised Power Transfer Function" |H(s)| 2 .
The result is that RH (RC) is true. (as done in previous papers, Fausto Galetto, 2014Galetto, , 2018.