For a sequence *a*_{n}, *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,… (*a*_{n}=1 for all *n*), then the Dirichlet series is the Riemann zeta function

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For the alternating sequence 1,-1,1,-1,… (*a*_{n}=(-1)^{n-1}), the Dirichlet series is the Dirichlet eta function

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Now, consider the case where *a*_{n}=*f*(*n*), where *f*(*n*) is a multiplicative function.

Then we have Dirichlet series generating function:

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Using the prime factorization of *n*, , we see:

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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:

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When *f*(*n*) is a completely multiplicative function, , so then the series within the summation becomes a geometric series:

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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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Tags: Dirichlet Series, Dirichlet Series Generating Function, Euler Product, Math, Monday Math, Multiplicative Function, Riemann Zeta Function

This entry was posted on August 31, 2009 at 12:18 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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