In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, a Dirichlet series is any
series of the form
where ''s'' is
complex, and
is a complex
sequence
In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is cal ...
. It is a special case of
general Dirichlet series.
Dirichlet series play a variety of important roles in
analytic number theory
In 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 Peter Gustav Lejeune Dirichlet's 1837 introduction of Dir ...
. The most usually seen definition of the
Riemann zeta function is a Dirichlet series, as are the
Dirichlet L-functions. Specifically, the Riemann zeta function ''ζ(s)'' is the Dirichlet series of the constant unit function ''u(n)'', namely:
where ''D(u, s)'' denotes the Dirichlet series of ''u(n)''.
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.
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 ''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 ''a
n'' 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
In mathematics, a set ''A'' is a subset of a set ''B'' if all elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subse ...
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 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 example of a Dirichlet series is
:
whose analytic continuation to
(apart from a simple pole at
) is the
Riemann zeta function.
Provided that is real-valued at all natural numbers , the respective real and imaginary parts of the Dirichlet series 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. It is this bit of combinatorics which inspires the
Euler product formula.
Another is:
:
where 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 is a
Dirichlet L-function.
If the
arithmetic function has a
Dirichlet inverse function
, i.e., if there exists an inverse function such that the Dirichlet convolution of ''f'' with its inverse yields the multiplicative identity
, then the
DGF of the inverse function is given by the reciprocal of ''F'':
:
Other identities include
:
where
is the
totient function,
:
where ''J
k'' is the
Jordan function, and
:
where ''σ''
''a''(''n'') is the
divisor function. By specialization to the divisor function ''d'' = ''σ''
0 we have
:
The logarithm of the zeta function is given by
:
where Λ(''n'') is the
von Mangoldt function
In mathematics, the von Mangoldt function is an arithmetic function named after German mathematician Hans von Mangoldt. It is an example of an important arithmetic function that is neither multiplicative nor additive.
Definition
The von Mang ...
. Similarly, we have that
:
The
logarithmic derivative of the zeta function is then
:
These last three are special cases of a more general relationship for derivatives of Dirichlet series, given below.
Given the
Liouville function ''λ''(''n''), one has
:
Yet another example involves
Ramanujan's sum:
:
Another pair of examples involves the
Möbius function and the
prime omega function:
:
:
We have that the Dirichlet series for the
prime zeta function, which is the analog to the
Riemann zeta function summed only over indices ''n'' which are prime, is given by a sum over the
Moebius function and the logarithms of the zeta function:
:
A large tabular catalog listing of other examples of sums corresponding to known Dirichlet series representations is foun
here
Examples of Dirichlet series DGFs corresponding to
additive (rather than multiplicative) ''f'' are given
here for the
prime omega functions
and
, which respectively count the number of distinct prime factors of ''n'' (with multiplicity or not). For example, the DGF of the first of these functions is expressed as the product of the
Riemann zeta function and the
prime zeta function for any complex ''s'' with
:
:
If ''f'' is a
multiplicative function
In number theory, a multiplicative function is an arithmetic function f of a positive integer n with the property that f(1)=1 and
f(ab) = f(a)f(b) whenever a and b are coprime.
An arithmetic function is said to be completely multiplicative (o ...
such that its DGF ''F'' converges absolutely for all
, and if ''p'' is any
prime number
A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime ...
, we have that
:
where
is the
Moebius function. Another unique Dirichlet series identity generates the summatory function of some arithmetic ''f'' evaluated at
GCD inputs given by
:
We also have a formula between the DGFs of two arithmetic functions ''f'' and ''g'' related by
Moebius inversion. In particular, if
, then by Moebius inversion we have that
. Hence, if ''F'' and ''G'' are the two respective DGFs of ''f'' and ''g'', then we can relate these two DGFs by the formulas:
:
There is a known formula for the exponential of a Dirichlet series. If
is the DGF of some arithmetic ''f'' with
, then the DGF ''G'' is expressed by the sum
:
where
is the
Dirichlet inverse of ''f'' and where the
arithmetic derivative of ''f'' is given by the formula
for all natural numbers
.
Analytic properties
Given a sequence
of complex numbers we try to consider the value of
:
as a function of the
complex
Complex commonly refers to:
* Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe
** Complex system, a system composed of many components which may interact with each ...
variable ''s''. In order for this to make sense, we need to consider the convergence properties of the above infinite series:
If
is a
bounded sequence of complex numbers, then the corresponding Dirichlet series ''f'' converges
absolutely on the open half-plane Re(''s'') > 1. In general, if ''a
n'' = O(''n
k''), the series converges absolutely in the half plane Re(''s'') > ''k'' + 1.
If the set of sums
:
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
In 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. Functions of each type are infinitely differentiable, but complex ...
on the corresponding open half plane.
In general
is the abscissa of convergence of a Dirichlet series if it converges for
and diverges for
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.
Abscissa of convergence
Suppose
:
converges for some
:Proposition 1.
''Proof.'' Note that:
:
and define
:
where
:
By
summation by parts we have
:
:Proposition 2. Define
::
:Then:
::
:is the abscissa of convergence of the Dirichlet series.
''Proof.'' From the definition
:
so that
:
which converges as
whenever
Hence, for every
such that
diverges, we have
and this finishes the proof.
:Proposition 3. If
converges then
as
and where it is meromorphic (
has no poles on
).
''Proof.'' Note that
:
and
we have by summation by parts, for
:
Now find ''N'' such that for ''n'' > ''N'',
:
and hence, for every
there is a
such that for
:
:
Formal Dirichlet series
A formal Dirichlet series over a ring ''R'' is associated to a function ''a'' from the positive integers to ''R''
:
with addition and multiplication defined by
:
:
where
:
is the
pointwise In mathematics, the qualifier pointwise is used to indicate that a certain property is defined by considering each value f(x) of some Function (mathematics), function f. An important class of pointwise concepts are the ''pointwise operations'', that ...
sum and
:
is the
Dirichlet convolution of ''a'' and ''b''.
The formal Dirichlet series form a ring Ω, indeed an ''R''-algebra, with the zero function as additive zero element and the function ''δ'' defined by ''δ''(1) = 1, ''δ''(''n'') = 0 for ''n'' > 1 as multiplicative identity. An element of this ring is invertible if ''a''(1) is invertible in ''R''. If ''R'' is commutative, so is Ω; if ''R'' is an
integral domain
In mathematics, an integral domain is a nonzero commutative ring in which the product of any two nonzero elements is nonzero. Integral domains are generalizations of the ring of integers and provide a natural setting for studying divisibilit ...
, so is Ω. The non-zero multiplicative functions form a subgroup of the group of units of Ω.
The ring of formal Dirichlet series over C is isomorphic to a ring of formal power series in countably many variables.
Derivatives
Given
:
it is possible to show that
:
assuming the right hand side converges. For a
completely multiplicative function
In number theory, functions of positive integers which respect products are important and are called completely multiplicative functions or totally multiplicative functions. A weaker condition is also important, respecting only products of coprime ...
ƒ(''n''), and assuming the series converges for Re(''s'') > σ
0, then one has that
:
converges for Re(''s'') > σ
0. Here, Λ(''n'') is the
von Mangoldt function
In mathematics, the von Mangoldt function is an arithmetic function named after German mathematician Hans von Mangoldt. It is an example of an important arithmetic function that is neither multiplicative nor additive.
Definition
The von Mang ...
.
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
:
Coefficient inversion (integral formula)
For all positive integers
, the function ''f'' at ''x'',
, can be recovered from the
Dirichlet generating function (DGF) ''F'' of ''f'' (or the Dirichlet series over ''f'') using the following integral formula whenever
, the
abscissa of absolute convergence of the DGF ''F''
:
It is also possible to invert the
Mellin transform of the summatory function of ''f'' that defines the DGF ''F'' of ''f'' to obtain the coefficients of the Dirichlet series (see section below). In this case, we arrive at a complex
contour integral formula related to
Perron's theorem. Practically speaking, the rates of convergence of the above formula as a function of ''T'' are variable, and if the Dirichlet series ''F'' is sensitive to sign changes as a slowly converging series, it may require very large ''T'' to approximate the coefficients of ''F'' using this formula without taking the formal limit.
Another variant of the previous formula stated in Apostol's book provides an integral formula for an alternate sum in the following form for
and any real
where we denote
:
:
Integral and series transformations
The
inverse Mellin transform of a Dirichlet series, divided by s, is given by
Perron's formula.
Additionally, if
is the (formal) ordinary
generating function of the sequence of
, then an integral representation for the Dirichlet series of the generating function sequence,
, is given by
:
Another class of related derivative and series-based
generating function transformations on the ordinary generating function of a sequence which effectively produces the left-hand-side expansion in the previous equation are respectively defined in.
Relation to power series
The sequence ''a
n'' generated by a Dirichlet series generating function corresponding to:
:
where ''ζ''(''s'') is the
Riemann zeta function, has the ordinary generating function:
:
Relation to the summatory function of an arithmetic function via Mellin transforms
If ''f'' is an
arithmetic function with corresponding DGF ''F'', and the summatory function of ''f'' is defined by
:
then we can express ''F'' by the
Mellin transform of the summatory function at
. Namely, we have that
:
For
and any natural numbers
, we also have the approximation to the DGF ''F'' of ''f'' given by
:
See also
*
General Dirichlet series
*
Zeta function regularization
In mathematics and theoretical physics, zeta function regularization is a type of regularization (physics), regularization or summability method that assigns finite values to Divergent series, divergent sums or products, and in particular can be ...
*
Euler product
*
Dirichlet convolution
References
*
*
The general theory of Dirichlet's series by G. H. Hardy. Cornell University Library Historical Math Monographs.
Cornell University Library Digital Collections*
*
*
*
{{Authority control
Zeta and L-functions
Series (mathematics)
Series expansions