Elementary Proof of Riemann’s Hypothesis by the Modified Chi-square Function

The present article shows a proof of Riemann’s Hypothesis (RH) which is both general (i.e. valid for all the non-trivial zeroes of the zeta function) and elementary (that is not using the theory of the complex functions) in which the real constant σ=+1/2 arises by itself and automatically. The modified chi-square function in one of its four forms (±1/∙)Xk 2 (Ω,x/ω) is used as an interpolating function of the progressions {n ±α }, of their summations {∑n ±α } and of the progressions {N ±α+1 /(±α+1)}, with α∈R n,N∈N so that k=2±2α and in the real plane (α,k) two half-lines are set up with k<2. The use of the Euler-MacLaurin formula with the oneto-one correspondence between the summation operation ∑ and the shift vector operator Σ≡(Σα,Σk) in the real 2D plane (α,k) lead to find the zeroes of Euler’s function. Finally, the extrusion to the third imaginary axis it leads to prove Riemann’s hypothesis. Key-Words: Riemann’s hypothesis, modified chi-square function, numeric progressions


Introduction
Since 1859, when formulated by G.F.B. Riemann for the first time, Riemann's hypothesis (RH), "all the non-trivial zeroes of the zeta function ζ(s)=ζ(σ+it) have real part equal to +1/2", has been a challenge in number theory in that, though accepted and experimentally verified up to values of t ≈ 10 15 and beyond, it has never been proven. Its proof would have important consequences in both mathematics and physics [1][2][3][4][5][6][7][8]. As the experimental confirmation is not enough for mathematics, the present article shows a general (that is valid for any value of t∈R up to ∞) and elementary (in the sense that it does not use the theory of complex functions) proof of RH. The proof uses some simple techniques [9][10][11]: 1) the modified chi-square function as an interpolating function of the progressions {n ±α }, of their summations {∑n ±α } and of the progressions {N ±α+1 /(±α+1)}, with α∈R n,N∈N and the relationships k=2±2α; 2) the Euler-MacLaurin formula valid for high enough N values; 3) the on-to-one correspondence between the summation operation ∑ and the shift vector operator Σ in the plane (a k) so that the Euler equation ζ(α)=0 can be replaced by Σ=0=null_vector i.e. │Σ│= 0 thus finding all the roots of Euler's equations that is the zeroes of the Euler zeta function; 4) the extrusion to the third imaginary axis it in both the positive and negative direction thus finding all the zeroes of Riemann's zeta functions, in the same way as above, all of them having real part +1/2.

Problem Formulation
The modified chi-square function (±1/•)X k 2 (Ω,x/ω) with k degrees of freedom in one of its four forms: (1) with k<2 is a general version of the standard chisquare function X k 2 (x) also used in statistics [12][13][14][15] with the values Ω = ω = 1 and is used as a fit function of the finite progressions {n ±α }, of their summations {∑ (n=1→N) n ±α } and of the progressions {N ±α+1 /(±α+1)} taking advantage of the adjustment of its parameters k, Ω and ω just like the f(x)=x ±α function and just as the Γ(x)=Γ x function is an interpolating function of factorial numbers n! being x∈R + and n∈N.
Both examples show that the decay (or growth according to the case) parameter is much greater than the number of terms x o >>n max as already anticipated.
However, it is of the utmost importance to highlight that the latest relation Δk = ±2 holds for α < -1 and α > 0 in that within the range α ∈ (-1,0) there is a different situation, as described in the following.
Going back again to Fig. 8 all the Σ vectors lay on the plane k=+1 along the strip σ∈(-1/2, +1/2) with their application points on (σ k it)≡(-1/2 1 it) and their end points on (σ k it)≡(+1/2 1 it) being all horizontal that is parallel to the complex plane s that is k=0.

Conclusion
The innovative methodology of fitting the numeric progressions {n ±α } as well as {∑n ±α } and {N ±α+1 /(±α+1)} by the modified chi-square function in one of its four forms (±1/•)X k 2 (Ω,x/ω) has led to an elementary proof of Riemann's hypothesis (elementary in the sense that it does not use the theory of complex functions), a results that has been awaited since 157 years ago and never attained before now.
It has to be remarked that, in this proof, the real constant σ = +1/2 arises by itself and automatically in a straightforward way.
As for the future developments, the first one is the study of this shift vector operator also in the light of Hilbert and Pòlya conjecture, while further topics will concern the use of the modified chi-square function with k<2 for the analytical treatment of the finite sequences of prime numbers, with the goal of getting a more refined version of the prime number theorem (PNT), as well as the use of the same function with k>2 for the statistical treatment of the normalized spacing of the non-trivial zeroes of Riemann's zeta function in the frame of random matrices and Gaussian ensembles.