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 |