The Riemann Hypothesis is false

13 February 2026, Version 1
This content is an early or alternative research output and has not been peer-reviewed by Cambridge University Press at the time of posting.

Abstract

Let $\Theta$ denote the supremum of the real parts of the zeros of the Riemann zeta function $\zeta(s)$. By carefully exploiting the Landau-Gonek explicit formula, the Dirichlet polynomial approximation of $\zeta(s)$ and other classical results in analytic number theory, we rigorously demonstrate that $\Theta=1$. This entails the existence of infinitely many Riemann zeros off the critical line (thus disproving the Riemann Hypothesis (RH)). The paper is concluded by a brief discussion of why our argument does not work for both Weil and Beurling zeta functions, whose analogues of the RH are known to be true.

Keywords

Riemann Hypothesis
false
disproof
Landau-Gonek explicit formula

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