Cook's Riemann Hypothesis Proof

Jeff Cook has his third Riemann Hypothesis Proof in preprint here and now on this site!

Why a third?

In short, straight-forward simplicity!

But there's more to it than just that. The first proof, while completely valid, was very abstract, perhaps too abstract for such an important mathematical problem. The second proof was simply added to the same document as the first, making the entire paper longer and in many cases too broad. Additionally, in the second proof, while less abstract, actually has a number of the equations incorrectly written, which leads to incorrect values, falsifying everything sadly. It was an irresponsible hurried job that caused so much bitterness for Mr. Cook that he refused to even revisit his notes to dig up the correctly written equations to match the results of the paper he intended to present..

But now time has passed, and Mr. Cook still aims to establish himself as an accomplished mathematician. He realized that this important problem, the Riemann Hypothesis, needs to be put to rest once and for all, and due to his love for analytical number theory, he is determined to be the one to do it.

He can only hope the third time's a charm.

What's in this paper is immensely more robust than anything mathematical he has put forth before. He speaks in the language of mathematics only, allowing equations, values, graphs and tables to do the talking rather than discussing such matters verbally. The entire paper is only 30 pages long, consists of just 6 lemmas, 5 of which are proven using only elementary arithmetic. He introduces no new mathematics and leaves little room for argument for what he shows. He simply allows his findings to be presented step by step, based on prior proven theorems, which is what a mathematical proof should consist of..

Note: at this time, the paper contains no references. As hinted above, the theorems used in his proof are so common and fundamental that he feels it is satisfactory at this pre-print stage to go out for others to begin hammering on without references, so long as it is understood that proper citations will be added soon. For comments or questions, please email the author directly.

The pre-print paper is now available as a PDF, 261 KB file.

Here it is:

Abstract:

A proof of the Riemann Hypothesis is proposed in six lemmas, where five of the six are proven using elementary arithmetic. It is shown that by applying all the zeros of the zeta function to a ratio, having an infinite number of numerators and divisors equal to the same value, the modulus of a variable z used to calculate the ratio are all equal to the square root of one divided by fourteen for all the zeros of zeta of s, trivial or non-trivial. Using the common modulus, it is shown that the value of the ratio for all the non-trivial zeros is a fixed constant, whereby allowing one to calculate the only possible positive Real part of s for the non-trivial zeros. Such proof suggests that the greatest common multiple and lowest common denominator of this ratio for all the zeros of zeta of s lie in the non-trivial zeros with a fixed Real part one half.

For the complete paper, click A Proof of the Riemann Hypothesis, by Jeffrey N. Cook, completed precisely on his 38th birthday, 11-12-2009, oddly enough. Happy Birthday, Mr. Cook!