The Riemann Hypothesis is false

TL;DR

The paper claims to prove that the supremum of the real parts of zeros of the Riemann zeta function is 1, thereby disproving the Riemann Hypothesis by showing infinitely many zeros off the critical line.

Contribution

It presents a proof that the Riemann Hypothesis is false by establishing that the supremum of zero real parts is 1, contrary to the traditional belief that it is 1/2.

Findings

Establishes that $ heta=1$, indicating zeros off the critical line.

Disproves the Riemann Hypothesis.

Provides a rigorous argument for the existence of infinitely many zeros with real part 1.

Abstract

Let $\Theta$ denote the supremum of the real parts of the zeros of the Riemann zeta function. We demonstrate that $\Theta=1$, which entails the existence of infinitely many Riemann zeros off the critical line (thus disproving the Riemann Hypothesis (RH), which asserts that $\Theta = \frac{1}{2}$). The paper is concluded by a brief discussion of why our argument doesn't work for both Weil and Beurling zeta functions whose analogues of the RH are known to be true. NB: The author believes that the paper is now clear and rigorous enough for someone with at least a graduate level of familirity with analytic number theory. Therefore, this shall be the very final revision. Addendum (11 February 2026): On page 4 of the previous version (version 47), there is a minor typo where the author wrote F''(x_0)=-v^{2}/T instead of F''(x_0) = -(v/T)^2. A correction of this typo reveals that the stationery phase approximation (SPA) gives a weak upper bound for J_{0}(T) when x_0 is small. The latest (and final) version circumvents the use of the SPA by directly summing J(T) over m, and taking the summation inside the integral (equation (20)). This allows us to invoke Van de Corput-type bounds, which are uniformly sharp for our purposes. The rest of the paper remains as it was.