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

Riemann’s hypothesis has always been a challenge in number theory, thus the present article shows a general (i.e. valid for any non-trivial zero) and elementary (i.e. not using the theory of complex functions) proof of it, in which the constant +1/2 arises by itself and automatically. In addition the method gives the values of all the trivial zeroes of Riemann’s function. The following steps are used: The modified chi-square function with its parameters Ω, k and ω=ω(k), in one of its four forms (±1∙/)Χk2(Ω,x/ω), as the interpolating function of the {nα} progressions and of the sums {∑nα} with α∈R so that k=2±2α for α<0 and α>0 respectively; the Euler-MacLaurin formula; the shift real vector operator Σ≡ (Σα Σk)≡ (Δα Δk)≡ (+1 (4α–2)) in the Euclidean 2D space (α k) and its extrusion to the imaginary axis it, leading to the 3D shift complex vector operator Σ≡(Σσ Σk Σit)≡(Δσ Δk iΔt)≡(+1 (4σ–2) it) with norm Σ2=16σ2–16σ+5+t2. The condition Σ=0 that is │Σ│=Σ=0 leads to prove RH.