AUTHOR(S): 

Daniele Lattanzi

TITLE

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

KEYWORDS

Riemann’s hypothesis, modified chi-square function, numeric progressions

ABSTRACT

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 s=+1/2 arises by itself and automatically. The modified chi-square function in one of its four forms (±1/·)Xk^2(O,x/?) is used as an interpolating function of the progressions {n±a}, of their summations {Sn±a} and of the progressions {N±a+1/(±a+1)}, with a?R n,N?N so that k=2±2a and in the real plane (a,k) two half-lines are set up with k<2. The use of the Euler-MacLaurin formula with the one-to-one correspondence between the summation operation S and the shift vector operator S=(Sa,Sk) in the real 2D plane (a,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.

Cite this paper

Daniele Lattanzi. (2017) Elementary Proof of Riemann’s Hypothesis by the Modified Chi-square Function. International Journal of Mathematical and Computational Methods, 2, 6-12