The Zeta Function and the Prime Number Theorem
Euler was aware of the connection between the zeta function
with
and the sequence of prime numbers 2, 3, 5, 7,...
To see this notices that
so that
and similarly
+...
Continuing indefinitely we have
so that
with(1)
The bracketed product includes all terms of the form
where
is a prime number. This has implications for the counting function
which counts all the prime numbers less than
and the Prime Number Theorem:
Riemann showed that a proof of the Prime Number Theorem could be given if certain properties of the zeta function could be established. This is now called the Riemann hypothesis. These properties concern the location of the zeros of the zeta function. Equation (1) shows that the zeta function has no zeros for
but Riemann found an analytic continuation of the zeta function to
and the this function did have zeros.
