In
mathematics, a Dirichlet series is any
series of the form
where ''s'' is
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 ...
, 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 calle ...
. 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 Diri ...
. The most usually seen definition of the
Riemann zeta function
The Riemann zeta function or Euler–Riemann zeta function, denoted by the Greek letter (zeta), is a mathematical function of a complex variable defined as \zeta(s) = \sum_^\infty \frac = \frac + \frac + \frac + \cdots for \operatorname(s) > ...
is a Dirichlet series, as are the
Dirichlet L-function
In mathematics, a Dirichlet ''L''-series is a function of the form
:L(s,\chi) = \sum_^\infty \frac.
where \chi is a Dirichlet character and ''s'' a complex variable with real part greater than 1. It is a special case of a Dirichlet series. B ...
s. It is conjectured that the
Selberg class
In mathematics, the Selberg class 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 co ...
of series obeys the
generalized Riemann hypothesis
The 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, whic ...
. The series is named in honor of
Peter Gustav Lejeune Dirichlet
Johann Peter Gustav Lejeune Dirichlet (; 13 February 1805 – 5 May 1859) was a German mathematician who made deep contributions to number theory (including creating the field of analytic number theory), and to the theory of Fourier series and ...
.
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
Fiber or fibre (from la, fibra, links=no) is a natural or artificial substance that is significantly longer than it is wide. Fibers are often used in the manufacture of other materials. The strongest engineering materials often incorporate ...
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 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
The Riemann zeta function or Euler–Riemann zeta function, denoted by the Greek letter (zeta), is a mathematical function of a complex variable defined as \zeta(s) = \sum_^\infty \frac = \frac + \frac + \frac + \cdots for \operatorname(s) > ...
.
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
The Möbius function is a multiplicative function in number theory introduced by the German mathematician August Ferdinand Möbius (also transliterated ''Moebius'') in 1832. It is ubiquitous in elementary and analytic number theory and most of ...
. This and many of the following series may be obtained by applying
Möbius inversion
Moebius, Möbius or Mobius may refer to:
People
* August Ferdinand Möbius (1790–1868), German mathematician and astronomer
* Theodor Möbius (1821–1890), German philologist
* Karl Möbius (1825–1908), German zoologist and ecologist
* Pa ...
and
Dirichlet convolution
In mathematics, the Dirichlet convolution is a binary operation defined for arithmetic functions; it is important in number theory. It was developed by Peter Gustav Lejeune Dirichlet.
Definition
If f , g : \mathbb\to\mathbb are two arithmetic fu ...
to known series. For example, given a
Dirichlet character
In analytic number theory and related branches of mathematics, a complex-valued arithmetic function \chi:\mathbb\rightarrow\mathbb is a Dirichlet character of modulus m (where m is a positive integer) if for all integers a and b:
:1) \ch ...
one has
:
where is a
Dirichlet L-function
In mathematics, a Dirichlet ''L''-series is a function of the form
:L(s,\chi) = \sum_^\infty \frac.
where \chi is a Dirichlet character and ''s'' a complex variable with real part greater than 1. It is a special case of a Dirichlet series. B ...
.
If the
arithmetic function has a
Dirichlet inverse
In mathematics, the Dirichlet convolution is a binary operation defined for arithmetic functions; it is important in number theory. It was developed by Peter Gustav Lejeune Dirichlet.
Definition
If f , g : \mathbb\to\mathbb are two arithmetic fun ...
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
In number theory, Euler's totient function counts the positive integers up to a given integer that are relatively prime to . It is written using the Greek letter phi as \varphi(n) or \phi(n), and may also be called Euler's phi function. In o ...
,
:
where ''J
k'' is the
Jordan function, and
:
where ''σ''
''a''(''n'') is the
divisor function
In mathematics, and specifically in number theory, a divisor function is an arithmetic function related to the divisors of an integer. When referred to as ''the'' divisor function, it counts the ''number of divisors of an integer'' (includin ...
. By specialization to the divisor function ''d'' = ''σ''
0 we have
:
The logarithm of the zeta function is given by
:
Similarly, we have that
:
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 Mango ...
. The
logarithmic derivative
In mathematics, specifically in calculus and complex analysis, the logarithmic derivative of a function ''f'' is defined by the formula
\frac
where f' is the derivative of ''f''. Intuitively, this is the infinitesimal relative change in ''f' ...
is then
:
These last three are special cases of a more general relationship for derivatives of Dirichlet series, given below.
Given the
Liouville function The Liouville Lambda function, denoted by λ(''n'') and named after Joseph Liouville, is an important arithmetic function.
Its value is +1 if ''n'' is the product of an even number of prime numbers, and −1 if it is the product of an odd number o ...
''λ''(''n''), one has
:
Yet another example involves
Ramanujan's sum
In number theory, Ramanujan's sum, usually denoted ''cq''(''n''), is a function of two positive integer variables ''q'' and ''n'' defined by the formula
: c_q(n) = \sum_ e^,
where (''a'', ''q'') = 1 means that ''a'' only takes on values coprime ...
:
:
Another pair of examples involves the
Möbius function
The Möbius function is a multiplicative function in number theory introduced by the German mathematician August Ferdinand Möbius (also transliterated ''Moebius'') in 1832. It is ubiquitous in elementary and analytic number theory and most of ...
and the
prime omega function In number theory, the prime omega functions \omega(n) and \Omega(n) count the number of prime factors of a natural number n. Thereby \omega(n) (little omega) counts each ''distinct'' prime factor, whereas the related function \Omega(n) (big omega) ...
:
:
:
We have that the Dirichlet series for the
prime zeta function In mathematics, the prime zeta function is an analogue of the Riemann zeta function, studied by . It is defined as the following infinite series, which converges for \Re(s) > 1:
:P(s)=\sum_ \frac=\frac+\frac+\frac+\frac+\frac+\cdots.
Properties
...
, which is the analog to the
Riemann zeta function
The Riemann zeta function or Euler–Riemann zeta function, denoted by the Greek letter (zeta), is a mathematical function of a complex variable defined as \zeta(s) = \sum_^\infty \frac = \frac + \frac + \frac + \cdots for \operatorname(s) > ...
summed only over indices ''n'' which are prime, is given by a sum over the
Moebius function
Moebius, Möbius or Mobius may refer to:
People
* August Ferdinand Möbius (1790–1868), German mathematician and astronomer
* Theodor Möbius (1821–1890), German philologist
* Karl Möbius (1825–1908), German zoologist and ecologist
* Pa ...
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
Additive may refer to:
Mathematics
* Additive function, a function in number theory
* Additive map, a function that preserves the addition operation
* Additive set-functionn see Sigma additivity
* Additive category, a preadditive category with fi ...
(rather than multiplicative) ''f'' are given
here
Here is an adverb that means "in, on, or at this place". It may also refer to:
Software
* Here Technologies, a mapping company
* Here WeGo (formerly Here Maps), a mobile app and map website by Here
Television
* Here TV (formerly "here!"), a ...
for the
prime omega function In number theory, the prime omega functions \omega(n) and \Omega(n) count the number of prime factors of a natural number n. Thereby \omega(n) (little omega) counts each ''distinct'' prime factor, whereas the related function \Omega(n) (big omega) ...
s
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
The Riemann zeta function or Euler–Riemann zeta function, denoted by the Greek letter (zeta), is a mathematical function of a complex variable defined as \zeta(s) = \sum_^\infty \frac = \frac + \frac + \frac + \cdots for \operatorname(s) > ...
and the
prime zeta function In mathematics, the prime zeta function is an analogue of the Riemann zeta function, studied by . It is defined as the following infinite series, which converges for \Re(s) > 1:
:P(s)=\sum_ \frac=\frac+\frac+\frac+\frac+\frac+\cdots.
Properties
...
for any complex ''s'' with
:
:
If ''f'' is a
multiplicative function
In number theory, a multiplicative function is an arithmetic function ''f''(''n'') 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 ''f''(''n'') is ...
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 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 because the only ways ...
, we have that
:
where
is the
Moebius function
Moebius, Möbius or Mobius may refer to:
People
* August Ferdinand Möbius (1790–1868), German mathematician and astronomer
* Theodor Möbius (1821–1890), German philologist
* Karl Möbius (1825–1908), German zoologist and ecologist
* Pa ...
. 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
In mathematics, the Dirichlet convolution is a binary operation defined for arithmetic functions; it is important in number theory. It was developed by Peter Gustav Lejeune Dirichlet.
Definition
If f , g : \mathbb\to\mathbb are two arithmetic fun ...
of ''f'' and where the
arithmetic derivative
In number theory, the Lagarias arithmetic derivative or number derivative is a function defined for integers, based on prime factorization, by analogy with the product rule for the derivative of a function that is used in mathematical analysis.
...
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
In mathematics, a function ''f'' defined on some set ''X'' with real or complex values is called bounded if the set of its values is bounded. In other words, there exists a real number ''M'' such that
:, f(x), \le M
for all ''x'' in ''X''. A ...
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
In mathematics, the radius of convergence of a power series is the radius of the largest disk at the center of the series in which the series converges. It is either a non-negative real number or \infty. When it is positive, the power series ...
for
power series
In mathematics, a power series (in one variable) is an infinite series of the form
\sum_^\infty a_n \left(x - c\right)^n = a_0 + a_1 (x - c) + a_2 (x - c)^2 + \dots
where ''an'' represents the coefficient of the ''n''th term and ''c'' is a const ...
. The Dirichlet series case is more complicated, though:
absolute convergence
In mathematics, an infinite series of numbers is said to converge absolutely (or to be absolutely convergent) if the sum of the absolute values of the summands is finite. More precisely, a real or complex series \textstyle\sum_^\infty a_n is sai ...
and
uniform convergence
In the mathematical field of analysis, uniform convergence is a mode of convergence of functions stronger than pointwise convergence. A sequence of functions (f_n) converges uniformly to a limiting function f on a set E if, given any arbitrarily ...
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
In mathematics, summation by parts transforms the summation of products of sequences into other summations, often simplifying the computation or (especially) estimation of certain types of sums. It is also called Abel's lemma or Abel transformati ...
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 f. An important class of pointwise concepts are the ''pointwise operations'', that is, operations defined ...
sum and
:
is the
Dirichlet convolution
In mathematics, the Dirichlet convolution is a binary operation defined for arithmetic functions; it is important in number theory. It was developed by Peter Gustav Lejeune Dirichlet.
Definition
If f , g : \mathbb\to\mathbb are two arithmetic fu ...
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, specifically abstract algebra, 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 se ...
, 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 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 Mango ...
.
Products
Suppose
:
and
:
If both ''F''(''s'') and ''G''(''s'') are
absolutely convergent
In mathematics, an infinite series of numbers is said to converge absolutely (or to be absolutely convergent) if the sum of the absolute values of the summands is finite. More precisely, a real or complex series \textstyle\sum_^\infty a_n is sa ...
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 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 In mathematics, the Mellin transform is an integral transform that may be regarded as the multiplicative version of the two-sided Laplace transform. This integral transform is closely connected to the theory of Dirichlet series, and is
often used ...
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 In mathematics, and more particularly in analytic number theory, Perron's formula is a formula due to Oskar Perron to calculate the sum of an arithmetic function, by means of an inverse Mellin transform.
Statement
Let \ be an arithmetic function, ...
.
Additionally, if
is the (formal) ordinary
generating function
In mathematics, a generating function is a way of encoding an infinite sequence of numbers () by treating them as the coefficients of a formal power series. This series is called the generating function of the sequence. Unlike an ordinary serie ...
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
The Riemann zeta function or Euler–Riemann zeta function, denoted by the Greek letter (zeta), is a mathematical function of a complex variable defined as \zeta(s) = \sum_^\infty \frac = \frac + \frac + \frac + \cdots for \operatorname(s) > ...
, 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 In mathematics, the Mellin transform is an integral transform that may be regarded as the multiplicative version of the two-sided Laplace transform. This integral transform is closely connected to the theory of Dirichlet series, and is
often used ...
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
Zeta (, ; uppercase Ζ, lowercase ζ; grc, ζῆτα, el, ζήτα, label=Demotic Greek, classical or ''zē̂ta''; ''zíta'') is the sixth letter of the Greek alphabet. In the system of Greek numerals, it has a value of 7. It was derived fr ...
*
Euler product In number theory, an Euler product is an expansion of a Dirichlet series into an infinite product indexed by prime numbers. The original such product was given for the sum of all positive integers raised to a certain power as proven by Leonhard E ...
*
Dirichlet convolution
In mathematics, the Dirichlet convolution is a binary operation defined for arithmetic functions; it is important in number theory. It was developed by Peter Gustav Lejeune Dirichlet.
Definition
If f , g : \mathbb\to\mathbb are two arithmetic fu ...
References
*
*
The general theory of Dirichlet's series by G. H. Hardy. Cornell University Library Historical Math Monographs.
Cornell University Library Digital Collections*
*
*
*
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Zeta and L-functions
Mathematical series
Series expansions