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Concept# Dirichlet series

Summary

In mathematics, a Dirichlet series is any series of the form
where s is complex, and is a complex sequence. It is a special case of general Dirichlet series.
Dirichlet series play a variety of important roles in analytic number theory. The most usually seen definition of the Riemann zeta function is a Dirichlet series, as are the Dirichlet L-functions. It is conjectured that the Selberg class of series obeys the generalized Riemann hypothesis. The series is named in honor of Peter Gustav Lejeune Dirichlet.
Dirichlet series can be used as generating series for counting weighted sets of objects with respect to a weight which is combined multiplicatively when taking Cartesian products.
Suppose that A is a set with a function w: A → N assigning a weight to each of the elements of A, and suppose additionally that the fibre over any natural number under that weight is a finite set. (We call such an arrangement (A,w) a weighted set.) Suppose additionally that an is the number of elements of A with weight n. Then we define the formal Dirichlet generating series for A with respect to w as follows:
Note that if A and B are disjoint subsets of some weighted set (U, w), then the Dirichlet series for their (disjoint) union is equal to the sum of their Dirichlet series:
Moreover, if (A, u) and (B, v) are two weighted sets, and we define a weight function w: A × B → N by
for all a in A and b in B, then we have the following decomposition for the Dirichlet series of the Cartesian product:
This follows ultimately from the simple fact that
The most famous example of a Dirichlet series is
whose analytic continuation to (apart from a simple pole at ) is the Riemann zeta function.
Provided that f is real-valued at all natural numbers n, the respective real and imaginary parts of the Dirichlet series F have known formulas where we write :
Treating these as formal Dirichlet series for the time being in order to be able to ignore matters of convergence, note that we have:
as each natural number has a unique multiplicative decomposition into powers of primes.

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