In mathematics, a Dirichlet series is any series of the form
where s is complex, and is a complex sequence. 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 Peter Gustav Lejeune Dirichlet.
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 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, if (A, u) and (B, v) are two weighted sets, and we define a weight function w: A × B → N 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
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 f is real-valued at all natural numbers n, the respective real and imaginary parts of the Dirichlet series F 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.
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 χ(n) one has
where L(χ, s) is a Dirichlet L-function.
If the arithmetic function f 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 Jk is the Jordan function, and
where σa(n) is the divisor function. By specialization to the divisor function d = σ0 we have
The logarithm of the zeta function is given by
Similarly, we have that
Here, Λ(n) is the von Mangoldt function. The logarithmic derivative 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 found 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 such that its DGF F converges absolutely for all , and if p is any prime number, 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 .
Given a sequence 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 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 an = O(nk), 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 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.
converges for some
Proof. Note that:
By summation by parts we have
Proof. From the definition
which converges as whenever Hence, for every such that diverges, we have and this finishes the proof.
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 :
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
is the pointwise 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, 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.
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, Λ(n) is the von Mangoldt function.
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
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 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 :
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.
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:
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
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
In mathematics, the classic Möbius inversion formula is a relation between pairs of arithmetic functions, each defined from the other by sums over divisors. It was introduced into number theory in 1832 by August Ferdinand Möbius.
In number theory, the prime number theorem (PNT) describes the asymptotic distribution of the prime numbers among the positive integers. It formalizes the intuitive idea that primes become less common as they become larger by precisely quantifying the rate at which this occurs. The theorem was proved independently by Jacques Hadamard and Charles Jean de la Vallée Poussin in 1896 using ideas introduced by Bernhard Riemann.
The Riemann zeta function or Euler–Riemann zeta function, ζ(s), is a mathematical function of a complex variable s, and can be expressed as:
In complex analysis, a branch of mathematics, analytic continuation is a technique to extend the domain of definition of a given analytic function. Analytic continuation often succeeds in defining further values of a function, for example in a new region where an infinite series representation in terms of which it is initially defined becomes divergent.
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 of primes.
In order theory, a field of mathematics, an incidence algebra is an associative algebra, defined for every locally finite partially ordered set and commutative ring with unity. Subalgebras called reduced incidence algebras give a natural construction of various types of generating functions used in combinatorics and number theory.
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.
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. It appears in a number of remarkable identities, including relationships on the Riemann zeta function and the Eisenstein series of modular forms. Divisor functions were studied by Ramanujan, who gave a number of important congruences and identities; these are treated separately in the article Ramanujan's sum.
In number theory, the Mertens function is defined for all positive integers n as
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.
In mathematics, the explicit formulae for L-functions are relations between sums over the complex number zeroes of an L-function and sums over prime powers, introduced by Riemann (1859) for the Riemann zeta function. Such explicit formulae have been applied also to questions on bounding the discriminant of an algebraic number field, and the conductor of a number field.
In mathematics, a natural number a is a unitary divisor of a number b if a is a divisor of b and if a and are coprime, having no common factor other than 1. Thus, 5 is a unitary divisor of 60, because 5 and have only 1 as a common factor, while 6 is a divisor but not a unitary divisor of 60, as 6 and have a common factor other than 1, namely 2. 1 is a unitary divisor of every natural number.
In mathematics, the prime zeta function is an analogue of the Riemann zeta function, studied by Glaisher (1891). It is defined as the following infinite series, which converges for :
In mathematics, the multiple zeta functions are generalisations of the Riemann zeta function, defined by
In number theory, an average order of an arithmetic function is some simpler or better-understood function which takes the same values "on average".
In mathematics, the Riemann hypothesis is a conjecture that the Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part 1/2. Many consider it to be the most important unsolved problem in pure mathematics. It is of great interest in number theory because it implies results about the distribution of prime numbers. It was proposed by Bernhard Riemann (1859), after whom it is named.
In the field of mathematical analysis, a general Dirichlet series is an infinite series that takes the form of
In number theory, the prime omega functions and count the number of prime factors of a natural number Thereby counts each distinct prime factor, whereas the related function counts the total number of prime factors of honoring their multiplicity. That is, if we have a prime factorization of of the form for distinct primes , then the respective prime omega functions are given by and . These prime factor counting functions have many important number theoretic relations.
In analytic number theory, a Dirichlet series, or Dirichlet generating function (DGF), of a sequence is a common way of understanding and summing arithmetic functions in a meaningful way. A little known, or at least often forgotten about, way of expressing formulas for arithmetic functions and their summatory functions is to perform an integral transform that inverts the operation of forming the DGF of a sequence. This inversion is analogous to performing an inverse Z-transform to the generating function of a sequence to express formulas for the series coefficients of a given ordinary generating function.
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