A Proof of the Riemann Hypothesis Using Bombieri's Equivalence Theorem

Abstract

The Riemann Hypothesis asserts that the Riemann $\xi(s)$ function has no zeros in the critical strip $0<{\rm Re}(s)<1$ except on the critical line ${\rm Re}(s)=\frac12$. Bombieri, in the official description of the Millennium Prize Problems, stated that the Riemann Hypothesis is equivalent to the condition that all local maxima of $\xi(t)$ on the critical line are positive and all local minima are negative. In this paper, we pursue this criterion. We first show that $\xi(s)$, when restricted to the critical line, satisfies a special differential equation, which ensures that it satisfies Bombieri's condition. Since a published proof of the sufficiency direction of Bombieri's theorem appears to be unavailable, we supply an independent proof of this implication. Using the Cauchy--Riemann equations, we prove that Bombieri's condition forces $\xi(s)$ to have no zeros off the critical line. The Riemann Hypothesis follows. We also discuss P\'olya's counterexample and other known counterexamples among L-functions, and demonstrate that none of them invalidates our approach.