Monday Math 88

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
.