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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.

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.

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.

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 Brief History of this Paper and its 2 Revisions 2 A Statement of the Problem and the General Approach to the Solution 3 Separating the Real and Imaginary Portions of the Complex Sum 4 Separating the Even and Odd Portions 5 Labeling the Functions 6 Separating the Sin and Cos Portions of the Real Even and Imaginary Even Sums 7 The Recursive Functional Relationships Between the Sums 8 Even Sums 9 The Conditions Generated by the Primary System of Sums and Constants 10 Requirements on the Sums of the System 11 A Brief Review of the System and the Dependency of Constants 12 The Odd Sum System 13 Combining the Requirements on the Constants 14 Conclusion Statement

# A Brief History of this Paper and its 2 Revisions

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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