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# Continue to: The Requirements on the Non-trivial Roots of the Riemann Zeta via the Dirichlet Eta Sum

The Riemann Hypothesis, known as a famous conjecture in mathematics, was proposed in 1859 by Bernhard Riemann in his paper "On the number of primes less than a given magnitude." The hypothesis makes a statement about the zeroes, or roots, of the similarly named Riemann Zeta function. Roots are complex numbers in this case, and it states that the only roots that exist, when the real portion of those numbers are greater than 0 and less than 1, are those where the real portion is exactly equal to 1/2.

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There are numerous resources on the function and hypothesis, how they relate to other problems in mathematics, of which there are many, and the title of this page is linked to Wikipedia for quick reference. This page is dedicated to a recent explanation of the hypothesis based on a closely related function that is commonly used in conjunction with the Zeta function, the Dirichlet Eta function.

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The proof is open to peer review. If you enjoy the solution, have questions, find errors or improvements, or wish to discuss or work on the material with the author, please contact me at the links provided on the Home or Info pages.

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There are 3 versions of this paper, an original and 2 revisions, with this being the 2nd update and 3rd version overall. The paper was put into digital text format around September of 2019, and the first revision around April of 2021. The original contains all core elements of the proof, but suffers from 2 issues. The reasoning of why certain functions must equal 0 is clumsy, as I was using the logical connections I had made at the time, compared to subsequently realized ones with greater efficiency, and I was still understanding the deeper reasons to such. Secondly, the system of equations that arises in the later half of the proof was not thoroughly explained, was redundant in some ways, and the key connections within it were explained using ratios that didn’t adequately shed light on the deeper logic of the system at the time.

By eighteen months later, I had gained more understanding of the system, and had made strides in explaining both the zeros in the first half of the proof and the system of equations in the second. The April 2021 revision made significant changes and improvements to the aforementioned issues, as well as to the editing, and to the clarity of the paper overall. However, and eventually, I thought the system had not yet revealed all of its secrets, nor had it been 100% explained, and that at its core, it must still contain a clearer explanation. After recently solidifying my understanding of certain connections within the system, and after two and a half years of pondering later, there are enough compelling changes to warrant an update, one which I think represents the clearest core logic and reasoning of the system to date. As such, this revision of the paper was put into digital format as of December, 2023.

Briefly, the major changes to the paper are as follows. One, an even clearer reasoning is provided as to why certain values must be 0. And two, the entire system of equations in the later half is revisited in a more orderly manner, organized, context added, and additional redundancies found and removed. At the time of updating, I suspect this will likely be the last revision to the paper, and I feel I’ve now fully explained what was set out to be explained regarding this approach to the topic.

A copy of the pdf can be downloaded via the pdf icon on the right, or on viXra here: https://vixra.org/pdf/1909.0515v3.pdf

A link to some formulae messing around with the connection between the Zeta and Inverse Fourier Transforms is included below.

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