The Riemann Zeta Function
We now cross from arithmetic into analysis. The Riemann zeta function is the
single object that ties calculus to the primes. It begins as an innocent-looking infinite sum, but
through the Euler product
it carries, encoded in its behaviour, the deepest secrets of how the primes are distributed.
The definition
\zeta(s) = \sum_{n=1}^{\infty} \frac{1}{n^{s}} = 1 + \frac{1}{2^{s}} + \frac{1}{3^{s}} + \frac{1}{4^{s}} + \cdots
For real s > 1 the series converges. As s
decreases toward 1 the value grows without bound — at
s = 1 it becomes the divergent harmonic series. Watch the curve climb to
a vertical asymptote at s = 1:
Famous values
Euler computed \zeta at the even integers, solving the celebrated
"Basel problem":
\zeta(2) = \frac{\pi^2}{6}, \qquad \zeta(4) = \frac{\pi^4}{90}.
That \pi appears in a sum over reciprocal squares is one of
mathematics' great surprises — a hint that \zeta reaches far beyond the
integers it is built from.
The leap to the complex plane
Riemann's revolutionary idea was to let s be a complex number,
and to extend \zeta by analytic continuation to (almost)
the whole plane — a single function defined even where the original sum diverges. This continued
\zeta(s) has "trivial zeros" at the negative even integers, and a family
of mysterious other zeros whose location is the central question of number theory — the
Riemann Hypothesis.
The bridge from this function to the primes is the
Euler product for zeta.