- Dirichlet series
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In mathematics, a Dirichlet series is any series of the form
where s and an are complex numbers and n = 1, 2, 3, ... . 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 Johann Peter Gustav Lejeune Dirichlet.
Contents
Combinatorial importance
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
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, and perhaps a bit more interestingly, if (A,u) and (B,v) are two weighted sets, and we define a weight function
by
- w(a,b) = u(a)v(b),
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
Examples
The most famous of Dirichlet series is
which is the Riemann zeta function.
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. It is this bit of combinatorics which inspires the Euler product formula.
Another is:
where μ(n) is the Möbius function. This and many of the following series may be obtained by applying Möbius inversion and Dirichlet convolution to known series. For example, given a Dirichlet character
one has
where L(χ,s) is a Dirichlet L-function.
Other identities include
where φ(n) is the totient function,
where Jk is the Jordan function, and
where σa(n) is the divisor function. By spezialiation to the divisor function d=σ0 follow
The logarithm of the zeta function is given by
for Re(s) > 1. Here,
is the von Mangoldt function. The logarithmic derivative is then
These last two are special cases of a more general relationship for derivatives of Dirichlet series, given below.
Given the Liouville function
, one has
Yet another example involves Ramanujan's sum:
Another example involves the Mobius function:
Analytic properties of Dirichlet series: the abscissa of convergence
Given a sequence {an}n ∈ N of complex numbers we try to consider the value of
as a function of the complex variable s. In order for this to make sense, we need to consider the convergence properties of the above infinite series:
If {an}n ∈ N is a bounded sequence of complex numbers, then the corresponding Dirichlet series f converges absolutely on the open half-plane of s such that Re(s) > 1. In general, if an = O(nk), the series converges absolutely in the half plane Re(s) > k + 1.
If the set of sums an + an + 1 + ... + an + k is bounded for n and k ≥ 0, then the above infinite series converges on the open half-plane of s such that Re(s) > 0.
In both cases f is an analytic function on the corresponding open half plane.
In general the abscissa of convergence of a Dirichlet series is the intercept on the real axis of the vertical line in the complex plane, such that there is convergence to the right of it, and divergence to the left. This is the analogue for Dirichlet series of the radius of convergence for power series. The Dirichlet series case is more complicated, though: absolute convergence and uniform convergence may occur in distinct half-planes.
In many cases, the analytic function associated with a Dirichlet series has an analytic extension to a larger domain.
Derivatives
Given
it is possible to show that
assuming the right hand side converges. For a completely multiplicative function ƒ(n), and assuming the series converges for Re(s) > σ0, then one has that
converges for Re(s) > σ0. Here,
is the von Mangoldt function.
Products
Suppose
and
If both F(s) and G(s) are absolutely convergent for s > a and s > b then we have
If a = b and ƒ(n) = g(n) we have
Integral transforms
The Mellin transform of a Dirichlet series is given by Perron's formula.
Relation to power series
The sequence an generated by a Dirichlet series generating function corresponding to:
where ζ(s) is the Riemann zeta function, has the ordinary generating function:
See also
References
- Apostol, Tom M. (1976), Introduction to analytic number theory, Undergraduate Texts in Mathematics, New York-Heidelberg: Springer-Verlag, ISBN 978-0-387-90163-3, MR0434929
- Hardy, G.H.; Riesz, Marcel (1915). The general theory of Dirichlet's series. Cambridge Tracts in Mathematics. 18. Cambridge University Press.
- The general theory of Dirichlet's series by G. H. Hardy. Cornell University Library Historical Math Monographs. {Reprinted by} Cornell University Library Digital Collections
- Gould, Henry W.; Shonhiwa, Temba (2008). "A catalogue of interesting Dirichlet series". Miss. J. Math. Sci. 20 (1). http://www.math-cs.ucmo.edu/~mjms/2008-1p.html.
- Mathar, Richard J. (2011). "Survey of Dirichlet series of multiplicative arithmetic functions". arXiv:1106.4038 [math.NT].
Categories:- Zeta and L-functions
- Mathematical series
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