The Fourier transform of the non-trivial zeros of the zeta function

TL;DR

This paper explores the Fourier transform of the non-trivial zeros of the Riemann zeta function, revealing connections to prime numbers and supporting the Hilbert-Polya conjecture related to the Riemann hypothesis.

Contribution

It introduces a novel analysis of the zeros and primes via Fourier transforms of a modified von Mangoldt function, providing evidence supporting the Riemann hypothesis.

Findings

Fourier transform reveals connections between zeta zeros and prime numbers.

Supports the Hilbert-Polya conjecture linking zeros to eigenvalues.

Provides computational evidence for the distribution of zeros in the critical line.

Abstract

The non-trivial zeros of the Riemann zeta function and the prime numbers can be plotted by a modified von Mangoldt function. The series of non-trivial zeta zeros and prime numbers can be given explicitly by superposition of harmonic waves. The Fourier transform of the modified von Mangoldt functions shows interesting connections between the series. The Hilbert-Polya conjecture predicts that the Riemann hypothesis is true, because the zeros of the zeta function correspond to eigenvalues of a positive operator and this idea encouraged to investigate the eigenvalues itself in a series. The Fourier transform computations is verifying the Riemann hypothesis and give evidence for additional conjecture that those zeros and prime numbers arranged in series that lie in the critical 1/2 positive upper half plane and over the positive integers, respectively.