In
mathematicsMathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, a
Dirichlet series is any
seriesA series is the sum of the terms of a sequence. Finite sequences and series have defined first and last terms, whereas infinite sequences and series continue indefinitely....
of the form
where
s and
an are
complex numberA complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...
s 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 theoryIn mathematics, analytic number theory is a branch of number theory that uses methods from mathematical analysis to solve problems about the integers. It is often said to have begun with Dirichlet's introduction of Dirichlet L-functions to give the first proof of Dirichlet's theorem on arithmetic...
. 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 classIn mathematics, the Selberg class S is an axiomatic definition of a class of L-functions. The members of the class are Dirichlet series which obey four axioms that seem to capture the essential properties satisfied by most functions that are commonly called L-functions or zeta functions...
of series obeys the
generalized Riemann hypothesisThe Riemann hypothesis is one of the most important conjectures in mathematics. It is a statement about the zeros of the Riemann zeta function. Various geometrical and arithmetical objects can be described by so-called global L-functions, which are formally similar to the Riemann zeta-function...
. The series is named in honor of
Johann Peter Gustav Lejeune DirichletJohann Peter Gustav Lejeune Dirichlet was a German mathematician with deep contributions to number theory , as well as to the theory of Fourier series and other topics in mathematical analysis; he is credited with being one of the first mathematicians to give the modern formal definition of a...
.
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
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
and
are two weighted sets, and we define a weight function
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
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 functionThe classical Möbius function μ is an important multiplicative function in number theory and combinatorics. The German mathematician August Ferdinand Möbius introduced it in 1832...
. This and many of the following series may be obtained by applying Möbius inversion and
Dirichlet convolutionIn mathematics, the Dirichlet convolution is a binary operation defined for arithmetic functions; it is important in number theory. It was developed by Johann Peter Gustav Lejeune Dirichlet, a German mathematician.-Definition:...
to known series. For example, given a
Dirichlet characterIn number theory, Dirichlet characters are certain arithmetic functions which arise from completely multiplicative characters on the units of \mathbb Z / k \mathbb Z...
one has
where
is a
Dirichlet L-function.
Other identities include
where φ(
n) is the totient function,
where J
k is the
Jordan functionIn number theory, Jordan's totient function J_k of a positive integer n is the number of k-tuples of positive integers all less than or equal to n that form a coprime -tuple together with n. This is a generalisation of Euler's totient function, which is J1...
, and
where σ
a(
n) is the
divisor functionIn mathematics, and specifically in number theory, a divisor function is an arithmetical function related to the divisors of an integer. When referred to as the divisor function, it counts the number of divisors of an integer. It appears in a number of remarkable identities, including relationships...
. By spezialiation to the divisor function
d=σ
0 follow
The logarithm of the zeta function is given by
for Re(
s) > 1. Here,
is the
von Mangoldt functionIn mathematics, the von Mangoldt function is an arithmetic function named after German mathematician Hans von Mangoldt.-Definition:The von Mangoldt function, conventionally written as Λ, is defined as...
. The
logarithmic derivativeIn mathematics, specifically in calculus and complex analysis, the logarithmic derivative of a function f is defined by the formulawhere f ′ is the derivative of f....
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 functionThe classical Möbius function μ is an important multiplicative function in number theory and combinatorics. The German mathematician August Ferdinand Möbius introduced it in 1832...
:
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
complexA complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...
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
absolutelyIn mathematics, a series of numbers is said to converge absolutely if the sum of the absolute value of the summand or integrand is finite...
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 functionIn mathematics, an analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions, categories that are similar in some ways, but different in others...
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 convergenceIn mathematics, the radius of convergence of a power series is a quantity, either a non-negative real number or ∞, that represents a domain in which the series will converge. Within the radius of convergence, a power series converges absolutely and uniformly on compacta as well...
for
power series. The Dirichlet series case is more complicated, though:
absolute convergenceIn mathematics, a series of numbers is said to converge absolutely if the sum of the absolute value of the summand or integrand is finite...
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 functionIn number theory, functions of positive integers which respect products are important and are called completely multiplicative functions or totally multiplicative functions. Especially in number theory, a weaker condition is also important, respecting only products of coprime numbers, and such...
ƒ(
n), and assuming the series converges for Re(
s) > σ
0, then one has that
converges for Re(
s) > σ
0. Here,
is the
von Mangoldt functionIn mathematics, the von Mangoldt function is an arithmetic function named after German mathematician Hans von Mangoldt.-Definition:The von Mangoldt function, conventionally written as Λ, is defined as...
.
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 transformIn mathematics, the Mellin transform is an integral transform that may be regarded as the multiplicative version of the two-sided Laplace transform...
of a Dirichlet series is given by
Perron's formulaIn mathematics, and more particularly in analytic number theory, Perron's formula is a formula due to Oskar Perron to calculate the sum of an arithmetical function, by means of an inverse Mellin transform.-Statement:...
.
Relation to power series
The sequence
generated by a Dirichlet series generating function corresponding to:
where
is the
Riemann zeta function, has the ordinary generating function:
