The Riemann zeta function has an infinity of zeros, that is, an infinity of values
of s for which ξ(s) = 0. In a paper presented to the Berlin Academy of Sciences
in 1859, Riemann showed all the important zeros were complex numbers that lay
in the critical strip bounded by x = 0 and x = 1. He also made his famous
hypothesis:
All the zeros of the Riemann zeta function ξ(s) lie on the line x = ½; the
line along the middle of the critical strip.
The first real step towards settling this hypothesis was made in 1896
independently by Charles de la Vallée-Poussin and Jacques Hadamard. They
showed that the zeros must lie on the interior of the strip (so x could not equal 0
or 1). In 1914, the English mathematician G.H. Hardy proved that an infinity of
zeros lie along the line x = ½ though this does not prevent there being an
infinity of zeros lying off it.
As far as numerical results go, the non-trivial zeros calculated by 1986
(1,500,000,000 of them) do lie on the line x = ½ while up-to-date calculations
have verified this is also true for the first 100 billion zeros. While these
experimental results suggest that the conjecture is reasonable, there is still the
possibility that it may be false. The conjecture is that all zeros lie on this critical
line, but this awaits proof or disproof.
Why is the Riemann hypothesis important?
There is an unexpected connection between the Riemann zeta function ξ (s)
and the theory of prime numbers (see page 36). The prime numbers are 2, 3, 5,