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 *p*_{n} is the *n*th 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.

### Like this:

Like Loading...

*Related*

Tags: Math, Monday Math, Prime Numbers, Riemann Zeta Function, Sieve of Eratosthenes

This entry was posted on February 16, 2009 at 12:29 am and is filed under Math/Science. You can follow any responses to this entry through the RSS 2.0 feed.
You can leave a response, or trackback from your own site.

February 23, 2009 at 12:06 am |

[…] and as , and for k>0, we have Now, via the geometric series, so and thus . Now, we showed previously that , and so . To find this last product, note that . Thus , and again using , we see that , and […]

November 25, 2010 at 10:42 am |

[…] that Thus, the series inside the prime product has only two non-zero terms, and , using the Euler product for the Riemann zeta function . We can similarly see: . Now, for a more challenging example, consider Euler’s […]