A Full and Detailed Proof for the Riemann Hypothesis & the Inductive Proof of Goldbach’s Conjecture

Abstract

As in my previous two papers [2] & [3] about the boundary of the prime gap still cause some misunderstanding, I here in this paper tries to clarify those detailed steps for the boundary of prime gap or a proof of Riemann Hypothesis. In a mirror image way (by the intuitive logic or the mathematical philosophy), the wrong assumption for s = 0.5 + vi in the Riemann Hypothesis equations (case III) may give us some well-known positive RH results and s acts as a counter-example to those case I & case II in a technical way. Indeed, all of the feasible case of the Riemann Zeta function with exponents from 1 to s = u + v*I become nonsense and imply RH is correct. For u = 0.5 with some real numbers v, these roots act as the counter examples of RH for the wrong assumption with no contradictions will be obtained. When the above proof and disproof work together with the undecidability from the Gödel’s incompleteness theorem, the Riemann Hypothesis is then correct. The truth of the hypothesis further implies that there is a need for the shift from the line x = 0 to 0.5 the zeta roots lie on it. NOT all of the points on x = 0.5 are zeros as the model equation in [2]. An application is in the quantum filtering for an elimination of noise without used for human counter-parts. This author suggests for the proof or disproof to any cases of hypothesis, one may need to find out those counter example(s) [14] among them. Actually, my proposition works very well for the cases in my disproof of Continuum Hypothesis [15] together with the proof in Riemann Hypothesis etc. In general, we may prove all similar statements that like the (Riemann) one by first find the counter-example for a disproof, the usual proof to the statement and the undecidability. Any contradictions will force out the proof to the wanted hypothesis statement.