A General Experimental Evidence of Riemann’s Hypothesis | Chapter 5 | Mathematics and Computer Science: Contemporary Developments Vol. 9

Riemann’s hypothesis has always been a challenge in number theory, thus the present article shows general (i.e. valid for any non-trivial zero from –∞ to +∞) and elementary (i.e. not using the theory of complex functions) experimental evidence of it, in which the constant +1/2 arises by itself and automatically. The following steps are used: the modified X-square function with its three parameters Ω, k and ω=ω(k), in one of its four forms (±1∙/)Χk²(Ω,x/ω), is used as the fit (that is interpolating) function of the {nα} progressions and of their sums {∑n^α} with α∈R so that k=2±2α for α<0 and α>0 respectively; the Euler-MacLaurin formula is implemented; the shift real vector operator Σ ≡ (Σ_α Σ_k)≡(Δα Δk)≡(+1 (4α–2)) in the Euclidean 2D space (α k) is used with its extrusion to the imaginary axis it, leading to the 3D shift complex vector operator Σ≡(Σ_σ Σ_k Σ_it)≡(Δσ Δk iΔt)≡(+1 (4σ–2)  it) with norm │Σ│=Σ=16σ²–16σ+5+t². Once applied the condition Σ=0 that is │Σ│=Σ=0 one gets σ1,2(t) = +1/2 ± i/4∙√(1+t²) ≈ +1/2 ± it/4  ∀t. This leads to validating R.H. for any non-trivial zero of the Zeta function. The methodology of experimental mathematics has been applied to fit i.e. to interpolate and extrapolate the numerical functions with all the statistical treatments and tools. The relevance of the work relies on the fact that, while waiting for a formal proof of R.H., it shows the validity of this hypothesis from an experimental viewpoint beyond any reasonable doubt.

 

Author (s) Details

 

Daniele Lattanzi
Nuclear Fusion Department, Frascati Research Centre, Frascati, Roma, Italy.

 

Please see the book here:- https://doi.org/10.9734/bpi/mcscd/v9/3263