The Zeta Function and the Prime Number Theorem

Euler was aware of the connection between the zeta function


and the sequence of prime numbers 2, 3, 5, 7,...

To see this notices that

so that

and similarly +...

Continuing indefinitely we have

so thatwith(1)

The bracketed product includes all terms of the formwhereis a prime number. This has implications for the counting functionwhich 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 forbut Riemann found an analytic continuation of the zeta function toand the this function did have zeros.