## Monday Math 59

Recall the Riemann zeta function
.

Now, consider the product . We see:
,
and we have the zeta function series with the even terms removed. Now, let us multiply the above by . Then we see:

where the remaining terms are those  where k is not divisible by 2 or 3. Multiplication of the above by  will eliminate those remaining terms divisible by 5; we may continue this procedure through the primes, and (in analogy with the Sieve of Eratosthenes) will, in the infinite product, eliminate every  term, k>1, giving
, where pn is the nth prime; thus the Riemann zeta function may be expressed as the product
.
Thus we see a connection between the primes and the Riemann zeta function. In fact, through this identity (first proved by Euler), the divergence of  at s=1 (divergence of the harmonic series) implies that there are infinitely many primes.