Number Theory

    

Geometric Symmetry of Non-Trivial Zeros of the Riemann Zeta Function in Polar Coordinates

Authors: Bryce Petofi Towne

This paper investigates the symmetry of non-trivial zeros of the Riemann zeta function (zeta(s)) through geometric analysis in polar coordinates. By transforming the complex number (s = sigma + it) into polar form, we demonstrate that the symmetry about the critical line (sigma = frac{1}{2}) necessitates (sigma = frac{1}{2}) for all non-trivial zeros. Numerical simulations further confirm the accuracy and consistency of this geometric approach. And we introduce a formula for the distribution pattern of all non-trivial zeros:[zetaleft(sqrt{frac{1}{4} + t^2} , e^{i arctan(2t)}ight) = 0]where: [r = sqrt{frac{1}{4} + t^2} quad text{and} quad theta = arctan(2t)]

Comments: 14 Pages.

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Submission history

[v1] 2024-08-09 08:47:16

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