Dirichlet series

Dirichlet series

Discussion

Encyclopedia
In mathematics
Mathematics
Mathematics 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 series
Series (mathematics)
A 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 number
Complex number
A 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 theory
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 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 class
Selberg class
In 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 hypothesis
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, which are formally similar to the Riemann zeta-function...

. The series is named in honor of Johann Peter Gustav Lejeune Dirichlet
Johann Peter Gustav Lejeune Dirichlet
Johann 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 function
Möbius function
The 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 convolution
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 Johann Peter Gustav Lejeune Dirichlet, a German mathematician.-Definition:...

to known series. For example, given a Dirichlet character
Dirichlet character
In 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 Jk is the Jordan function
Jordan's totient function
In 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 σa(n) is the divisor function
Divisor function
In 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 d0 follow

The logarithm of the zeta function is given by

for Re(s) > 1. Here, is the von Mangoldt function
Von Mangoldt function
In 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 derivative
Logarithmic derivative
In 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 function
Möbius function
The 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}nN of complex numbers we try to consider the value of

as a function of the complex
Complex number
A 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}nN is a bounded sequence of complex numbers, then the corresponding Dirichlet series f converges absolutely
Absolute convergence
In 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 function
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, 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 convergence
In 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 convergence
Absolute convergence
In 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 function
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. 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) > σ0, then one has that

converges for Re(s) > σ0. Here, is the von Mangoldt function
Von Mangoldt function
In 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 transform
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...

of a Dirichlet series is given by Perron's formula
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 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: