Last week, I demonstrated how the Dirichlet series generating function for a sequence consisting of a multiplicative function can be represented as a product over primes:

.

Now, let us use this for a couple of multiplicative functions. First, the Möbius function *μ*(*n*). We recall 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 totient function *φ*(*n*). It is a multiplicative function, and I showed here that for prime *p* and positive *k*, so the Dirichlet series generating function will have prime product form:

,

as when *k*=0, .

Now, to find a finite form for the series , we note that we can factor out (*p*-1) from all terms, and with a little work, we can get a geometric series:

.

Thus

,

and so we find the product over primes is

.

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Tags: Dirichlet Series, Dirichlet Series Generating Function, Euler Product, Math, Möbius Function, Monday Math, Prime Numbers, Totient

This entry was posted on September 7, 2009 at 12:01 am and is filed under Math/Science. You can follow any responses to this entry through the RSS 2.0 feed.
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October 26, 2009 at 12:19 am |

[…] which have Dirichlet series generating functions and , respectively. You may also recall that here I showed that the Dirichlet series generating function for |μ| is . I also demonstrated here […]