Monday Math 87

For a sequence an, n=1,2,3,…, the series is known as the Dirichlet series generating function (or sometimes simply the Dirichlet series) for the sequence. For example, if we take the trivial sequence 1,1,1,… (an=1 for all n), then the Dirichlet series is the Riemann zeta function
.
For the alternating sequence 1,-1,1,-1,… (an=(-1)n-1), the Dirichlet series is the Dirichlet eta function
.

Now, consider the case where an=f(n), where f(n) is a multiplicative function.
Then we have Dirichlet series generating function:
.
Using the prime factorization of n, , we see:
.
Now, we procede as we did in this post, which is just the above with multiplicative function . The above can be expressed as a product over the primes:
.
When f(n) is a completely multiplicative function, , so then the series within the summation becomes a geometric series:
.
Note that if our function is the constant function f(n)=1 (it is completely multiplicative), then the above gives us the Euler product for the Riemann zeta function

which we found here.

Note that even if the function is just multiplicative, and not completely multiplicative, it may still be possible to find a non-series expression for the series found in the product over primes. One example can be seen this past post.

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3 Responses to “Monday Math 87”

  1. Monday Math 88 « Twisted One 151’s Weblog Says:

    […] Math 88 By twistedone151 Last week, I demonstrated how the Dirichlet series generating function for a sequence consisting of a […]

  2. Monday Math 89 « Twisted One 151’s Weblog Says:

    […] , we get . [Note that , so that the x=0 case is the limit of the x≠0 general case]. Recall previouly that the Dirichlet series generating function for a sequence consisting of a multiplicative […]

  3. Monday Math 94 « Twisted One 151's Weblog Says:

    […] we can see that λ(n) has Dirichlet series generating function ; we can confirm this via the Euler product for Dirichlet series generating functions of completely multiplicative functions. One should also note that if a function f(n) has Dirichlet series generating function F(s), then […]

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