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The Riemann Hypothesis

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 attempted solution 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 solution 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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*As of 4/2/21 a new version of the proof is in the works. The current version, version 2 was found by the author to be incomplete, or inadequate in rigor, towards the end of the proof, and right for the wrong reason. The planned update addresses these issues as well as removing a redundancy around equation 51 in the proof. I hope to have the update in the next week or so.

The Requirements on the Non-trivial Roots of the Riemann Zeta via the Dirichlet Eta Sum

Use the table to jump to a specific page, or use the page navigation to browse each page here.

 1 A statement of the problem 2 Separating the Real and Imaginary Portions 3 Continuing Complex Separation 4 Separating the Even and Odd Portions 5 Separating the Sin and Cos Portions 6 Recursive Relationship Between Sums 7 More on the Recursive Relationship 8 Showing that the Sums must be 0 9 The System of Sums 10 More on the System of Sums 11 Solving the System of Sums 12 Conclusion Statement

A statement of the problem

The first section states the hypothesis in terms of the Dirichlet Eta sum. It is known that if you show the real portion of the input is 1/2, for all roots of the Eta sum in the given domain, then the same holds for the Zeta function.

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