Riemann Hypothesis

Prove that all non-trivial zeros of the Riemann zeta function ζ(s) lie on the critical line Re(s) = 1/2.

Background

The paper recalls the classical Riemann Hypothesis (RH) as the central problem motivating the development of a new variational framework for relating zeros of the core function Z0(t)=cos(θ(t)) to zeros of the Hardy Z-function Z(t).

Throughout the work, the author explores Edwards’ speculation and proposes a high-dimensional variational approach via sections ZN(t; ā), recasting RH as an optimization problem about maintaining reality of zeros along paths in parameter space.

References

All non-trivial zeros of $\zeta(s)$ lie on the critical line $Re(s) = 1/2$.

— On Edwards' Speculation and a New Variational Method for the Zeros of the $Z$-Function  (2405.12657 - Jerby, 2024) in Conjecture (RH), Introduction

All nontrivial zeros of the Riemann zeta function lie on the line Re s=\tfrac12. Equivalently, by Theorem~\ref{thm:tauberian_rh}, \alpha(\Phi)=\tfrac12.

— Regular Arithmetic Functions, Volume I. Theory, Applications, Examples  (2609.09366 - Cloitre, 8 Sep 2026) in Conjecture 3.6, Section 3.2

Indeed, Riemann assumed in 1859 that all nontrivial zeroes are placed on the line $Re(s)=\frac{1}{2}$. This is the famous Riemann's hypothesis. This problem has attracted considerable interest from many mathematicians, although after 163 years, it remains unsolved.

— The Generalized Riemann Zeta heat flow  (2402.10154 - Castillo et al., 2024) in Section 1.1 (Setting)

Despite extensive numerical evidence, the RH remains unproven.

— The Riemann Hypothesis Emerges in Dynamical Quantum Phase Transitions  (2511.11199 - Wei et al., 14 Nov 2025) in Introduction

One of the most famous open questions in mathematics is the Riemann Hypothesis, concerning the zeros of the Riemann zeta function.

— Automorphic Dirac Operators and an Analogue of the Quantum Harmonic Oscillator on $Γ\backslash \mathfrak{H}$  (2610.22291 - Manivel, 15 Oct 2026) in Section 1, Introduction

Riemann’s 1859 statement [21] on the non-trivial zeros of the zeta function, according to which it is “sehr wahrscheinlich” that all lie on the critical line, became known as the Riemann hypothesis.

— More than 8/9 zeros of the Riemann zeta function are simple or on the critical line  (2610.23467 - Cobeli et al., 16 Oct 2026) in Section 1, Introduction, page 1