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

TL;DR

This paper claims to prove the Riemann Hypothesis by establishing that Bombieri's equivalence theorem, combined with a differential equation satisfied by the zeta function on the critical line, implies no zeros outside this line.

Contribution

The paper provides an independent proof of Bombieri's equivalence theorem and applies it to prove the Riemann Hypothesis, which is a novel approach.

Findings

Proves that $\xi(s)$ satisfies a special differential equation on the critical line.

Establishes that Bombieri's equivalence condition implies no zeros outside the critical line.

Validates the proof despite Pólya's counterexample.

Abstract

The Riemann Hypothesis states that within the strip region of the complex plane $0 < {\rm Re}(s) < 1$, the Riemann $\xi(s)$ function has zeros only on the critical line ${\rm Re}(s) = \frac{1}{2}$ and none elsewhere. To prove the Riemann Hypothesis, we need to identify which points $s$ make the complex function $\xi(s) = 0$, which is evidently a challenging task. Bombieri proposed a proposition in the official description of the Millennium Prize Problems stating that "The Riemann hypothesis is equivalent to the statement that all local maxima of $\xi(t)$ (on the critical line) are positive and all local minima are negative." This provides a direction for proving the Riemann Hypothesis. In this paper, we follow Bombieri's approach to study the Riemann Hypothesis. First, we prove that the function $\xi(s)$ on the critical line (where it is a real function of a single real variable) satisfies a special differential equation. This ensures that it meets Bombieri's equivalence condition. Then, since we have not been able to find the original proof of Bombieri's equivalence theorem, we provide an independent proof for the sufficiency part of the theorem. We find that if the function $\xi(s)$ on the critical line satisfies Bombieri's equivalence condition, then by applying the Cauchy-Riemann equations to $\xi(s)$, we can prove that it has no zeros outside this critical line. Therefore, we can conclude that the Riemann Hypothesis is true. To further validate our findings, we discuss P\'olya's counterexample, which is misleading for understanding the research methods and results of this paper. However, we demonstrate that this counterexample does not invalidate Bombieri's equivalence theorem or the judgments proposed in this paper, nor does it affect our proof.