The Riemann Hypothesis is false

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.