General Mathematics


Riemann’s Functional Equation is Not a Valid Function and Its Implication on the Riemann Hypothesis

Authors: Armando M. Evangelista Jr.

Riemann’s functional equation was formulated by Riemann himself in order to extend the domain of the zeta function from the right half-plane into the entire complex plane except at s = 1. It also lead him to find a real function, so that, at s = ½ + ωi, the real function has zeros for some values of ω. Now, the real function was also related to the zeta function, which in turn has something to do with the distribution of prime numbers. This drove him to developed a formula of relating the zeros of the zeta function to the number of primes given a certain number. Riemann then conjectured that all the zeros of the zeta function are at s = ½ + ωi, which is now known as the Riemann Hypothesis. Hence, Riemann’s functional equation is the foundation upon which the Riemann Hypothesis is based. But, there is one problem, the function as shall be shown here, suffers from not being able to yield meaningful or valid values, as it should. Also, if one carefully examine on how Riemann arrived at his formula, I for one, found it to be unsatisfactory or unconvincing. It is, therefore, the aim of this present work to show, that, if carefully examined, Riemann’s functional equation could not be a valid function, and consequently, the Riemann Hypothesis crumbles on its claim.

Comments: 10 Pages.

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Submission history

[v1] 2018-08-28 11:37:38

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