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
.

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One Response to “Monday Math 88”

  1. Monday Math 94 « Twisted One 151’s Weblog Says:

    […] 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 […]

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