Number Theory

    

Integer Encoding Symmetries of the Riemann Zeta Function

Authors: Ryan Hackbarth

Here I present a derivation of an equation whose solution sets are the trivial and nontrivial zeros of the Riemann Zeta Function. I demonstrate how the trivial solutions are directly encoded by integer inputs and how these can be mapped by a symmetry to positive odd integers. I extend this insight to encode the even integers, and map these to the negative odd integers, which provides an explicit connection between particular values of the Riemann Zeta Function which have historical and ongoing research interest. I then extend this symmetry to the nontrivial zeroes, and demonstrate the dependence of the critical line in producing this symmetry. Finally, I note the distribution of the nontrivial zeroes have a correspondence with the distribution of trivial zeroes, and provide a first order approximation of this correspondence.

Comments: 11 Pages. This update includes a more usable primary function, and uses it to forecast the nontrivial zeroes of the Riemann Zeta Function.

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Submission history

[v1] 2026-01-21 06:33:14
[v2] 2026-01-27 04:37:39

Unique-IP document downloads: 191 times

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