A General Experimental Evidence of Riemann’s Hypothesis

Abstract

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 \(-\infty\) to \(+\infty\) ) 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 \(\Omega, \mathrm{k}\) and \(\omega=\omega(\mathrm{k})\), in one of its four forms \(( \pm 1 \cdot /) \mathrm{X}_{\mathrm{k}}{ }^2(\Omega, \mathrm{x} / \omega)\), is used as the fit (that is interpolating) function of the \(\left\{\mathrm{n}^\alpha\right\}\) progressions and of their sums \(\left\{\sum \mathrm{n}^\alpha\right\}\) with \(\alpha \in \mathrm{R}\) so that \(\mathrm{k}=2 \pm 2 \alpha\) for \(\alpha<0\) and \(\alpha>0\) respectively; the Euler-MacLaurin formula is implemented; the shift real vector operator \(\underline{\boldsymbol{\Sigma}} \equiv\left(\Sigma_\alpha \Sigma_{\mathrm{k}}\right) \equiv(\Delta \alpha \Delta \mathrm{k}) \equiv(+1(4 \alpha-2))\) in the Euclidean 2D space \((\underline{\boldsymbol{\alpha}} \underline{\mathbf{k}})\) is used with its extrusion to the imaginary axis i\(\underline{\boldsymbol{t}}\), leading to the 3D shift complex vector operator \(\underline{\boldsymbol{\Sigma}} \equiv\left(\Sigma_\sigma\right.\) \(\left.\Sigma_{\mathrm{k}} \Sigma_{\mathrm{it}}\right) \equiv(\Delta \sigma \Delta \mathrm{ki} \Delta \mathrm{t}) \equiv(+1(4 \sigma-2) \mathrm{it})\) with norm \(|\underline{\Sigma}|=\Sigma=16 \sigma^2-16 \sigma+5+\mathrm{t}^2\). Once applied the condition \(\underline{\Sigma}=\underline{0}\) that is \(|\underline{\Sigma}|=\Sigma=0\) one gets \(\sigma_{1,2}(\mathrm{t})=+1 / 2 \pm \mathrm{i} / 4 \cdot \sqrt{ }\left(1+\mathrm{t}^2\right) \approx+1 / 2 \pm \mathrm{it} / 4 \quad \forall \mathrm{t}\). 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.