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

Abstract

The Riemann hypothesis states that the Riemann xi(s) function has no zeros in the strip 0< Re(s)<1 except for the critical line where Re(s) = 1/2. However, determining whether a point in the strip is a zero of a complex function is a challenging task. Bombieri proposed an assertion in the Official Problem Description that states, ''The Riemann hypothesis is equivalent to the statement that all local maxima of xi(t) (namely, xi(s) on the critical line) are positive and all local minima are negative." In this paper, we follow Bombieri's idea to study the Riemann hypothesis. First, we show that on the critical line, the xi(s) function (which is then a real function of a single real variable) obeys a special differential equation, such that it satisfies Bombieri's equivalence condition. Then, since we have been unable to locate the original proof of Bombieri's equivalence theorem, we independently provide a proof for the sufficiency part of it. Namely, if the xi(s) function satisfies Bombieri's equivalence condition on the critical line, then by the Cauchy-Riemann equations, it has no zeros outside this critical line. Thus, we can conclude that the Riemann hypothesis is true. Finally, we comment on Pólya's counterexample against using the simplified xi(s) function to study the Riemann hypothesis, noting that it violates Bombieri's equivalence theorem.