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The Riemann hypothesis states that the Riemann $\xi(s)$ function has no zeros
in the strip $0< {\rm Re}(s) < 1$ except for the critical line where ${\rm
Re}(s) = 1/2$. 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 P\'olya's counterexample against using the $\xi(s)$
function to study the Riemann hypothesis, noting that it violates Bombieri's
equivalence theorem.