Riemann Hypothesis on the critical line

Prove that all nontrivial zeros of the Riemann zeta function ζ(s) lie on the critical line Re(s) = 1/2 in the complex plane.

Background

The paper investigates chaotic dynamics derived from the Riemann–von Mangoldt formula and proposes a chaotic operator that the authors argue is compatible with the Hilbert–Pólya perspective. They provide numerical observations (e.g., predominantly real eigenvalues from a random matrix model derived from their operator) that they claim support the Riemann Hypothesis.

Within this context, the Riemann Hypothesis is explicitly recalled as the central open conjecture about the location of nontrivial zeros of ζ(s), serving as the primary mathematical target that motivates the construction and analysis of their operator.

References

It conjectures that all nontrivial zeros of the Riemann zeta function lie on the critical line \text{Re}(s) = \frac{1}{2} , where s = \sigma + it is a complex number with real part \sigma and imaginary part t .

— If our chaotic operator is derived correctly, then the Riemann hypothesis holds true  (2404.00583 - Rafik, 2024) in Introduction (Section 1)

The Riemann hypothesis (RH) is one of the central open problems in mathematics and one of the seven Millennium Prize problems. This conjecture, which was formulated by Riemann in 1859, asserts that all the non-trivial zeros of the Riemann zeta function lie on the critical line $Re(s)=1/2$.

— A new proof that more than $2/3$ of the zeros of the Riemann zeta function are simple and on the critical line  (2609.02882 - Lamzouri, 2 Sep 2026) in Section 1, Introduction