Papers › A Proof of the Riemann Hypothesis Using Bombieri's Equivalence Theorem
A Proof of the Riemann Hypothesis Using Bombieri's Equivalence Theorem
Xiao Lin
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The Riemann Hypothesis asserts that the Riemann ξ(s) function has no zeros in the critical strip 0<Re(s)<1 except on the critical line Re(s)=1/2. 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 ξ(t) on the critical line are positive and all local minima are negative. In this paper, we pursue this criterion. We first show that ξ(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 ξ(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.
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