Dirichlet Series
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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 \sum_^\infty \frac, where ''s'' is complex, and a_n 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: \zeta(s) = \sum_^\infty \frac = \sum_^\infty \frac = D(u, s), 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 ''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: :\mathfrak^A_w(s) = \sum_ \frac 1 = \sum_^\infty \frac 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: :\mathfrak^_w(s) = \mathfrak^A_w(s) + \mathfrak^B_w(s). Moreover, if (''A'', ''u'') and (''B'', ''v'') are two weighted sets, and we define a weight function by :w(a,b) = u(a) v(b), for all ''a'' in ''A'' and ''b'' in ''B'', then we have the following decomposition for the Dirichlet series of the Cartesian product: :\mathfrak^_w(s) = \mathfrak^_u(s) \cdot \mathfrak^_v(s). This follows ultimately from the simple fact that n^ \cdot m^ = (nm)^.


Examples

The most famous example of a Dirichlet series is :\zeta(s)=\sum_^\infty \frac 1 , whose analytic continuation to \Complex (apart from a simple pole at s = 1) 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 s \equiv \sigma + i t: :\begin \Re (s)& = \sum_ \frac \\ \Im (s)& = -\sum_ \frac\,. \end Treating these as formal Dirichlet series for the time being in order to be able to ignore matters of convergence, note that we have: :\begin \zeta(s) &= \mathfrak^_(s) = \prod_ \mathfrak^_(s) = \prod_ \sum_ \mathfrak^_(s) \\ &= \prod_ \sum_ \frac = \prod_ \sum_ \left(\frac\right)^n = \prod_ \frac \end 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: :\frac=\sum_^\infty \frac 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 :\frac 1 =\sum_^\infty \frac where is a Dirichlet L-function. If the arithmetic function has a Dirichlet inverse function f^(n), i.e., if there exists an inverse function such that the Dirichlet convolution of ''f'' with its inverse yields the multiplicative identity \sum_ f(d) f^(n/d) = \delta_, then the DGF of the inverse function is given by the reciprocal of ''F'': :\sum_ \frac = \left(\sum_ \frac\right)^. Other identities include :\frac=\sum_^ \frac where \varphi(n) is the totient function, :\frac = \sum_^\infty \frac where ''Jk'' is the Jordan function, and :\begin & \zeta(s) \zeta(s-a)=\sum_^\infty \frac \\ pt& \frac = \sum_^\infty \frac \\ pt& \frac = \sum_^\infty \frac \end where ''σ''''a''(''n'') is the divisor function. By specialization to the divisor function ''d'' = ''σ''0 we have :\begin \zeta^2(s) & =\sum_^\infty \frac \\ pt\frac & =\sum_^\infty \frac \\ pt\frac & =\sum_^\infty \frac. \end The logarithm of the zeta function is given by :\log \zeta(s)=\sum_^\infty \frac\frac, \qquad \Re(s) > 1 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 :-\zeta'(s) = \sum_^ \frac, \qquad \Re(s) > 1. The logarithmic derivative of the zeta function is then :\frac = -\sum_^\infty \frac. 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 :\frac = \sum_^\infty \frac. Yet another example involves Ramanujan's sum: :\frac=\sum_^\infty\frac. Another pair of examples involves the Möbius function and the prime omega function: :\frac = \sum_^\infty \frac \equiv \sum_^\infty \frac. :\frac = \sum_^\infty \frac. 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: :P(s) := \sum_ p^ = \sum_ \frac \log \zeta(ns). 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 \omega(n) and \Omega(n), 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 \Re(s) > 1: :\sum_ \frac = \zeta(s) \cdot P(s), \Re(s) > 1. 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 \Re(s) > \sigma_, 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 :\left(1+f(p) p^\right) \times \sum_ \frac = \left(1-f(p) p^\right) \times \sum_ \frac, \forall \Re(s) > \sigma_, where \mu(n) is the Moebius function. Another unique Dirichlet series identity generates the summatory function of some arithmetic ''f'' evaluated at GCD inputs given by :\sum_ \left(\sum_^n f(\gcd(k, n))\right) \frac = \frac \times \sum_ \frac, \forall \Re(s) > \sigma_ + 1. We also have a formula between the DGFs of two arithmetic functions ''f'' and ''g'' related by Moebius inversion. In particular, if g(n) = (f \ast 1)(n), then by Moebius inversion we have that f(n) = (g \ast \mu)(n). Hence, if ''F'' and ''G'' are the two respective DGFs of ''f'' and ''g'', then we can relate these two DGFs by the formulas: :F(s) = \frac, \Re(s) > \max(\sigma_, \sigma_). There is a known formula for the exponential of a Dirichlet series. If F(s) = \exp(G(s)) is the DGF of some arithmetic ''f'' with f(1) \neq 0, then the DGF ''G'' is expressed by the sum :G(s) = \log(f(1)) + \sum_ \frac, where f^(n) is the Dirichlet inverse of ''f'' and where the arithmetic derivative of ''f'' is given by the formula f^(n) = \log(n) \cdot f(n) for all natural numbers n \geq 2.


Analytic properties

Given a sequence \_ of complex numbers we try to consider the value of : f(s) = \sum_^\infty \frac 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 ''an'' = O(''nk''), the series converges absolutely in the half plane Re(''s'') > ''k'' + 1. If the set of sums :a_n + a_ +\cdots + a_ 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 \sigma is the abscissa of convergence of a Dirichlet series if it converges for \Re(s) > \sigma and diverges for \Re(s) < \sigma. 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 :\sum_^\infty \frac converges for some s_0 \in \Complex, \Re(s_0) > 0. :Proposition 1. A(N) := \sum_^N a_n = o(N^). ''Proof.'' Note that: :(n+1)^s-n^s =\int_n^ s x^ \, dx = \mathcal(n^). and define :B(N) = \sum_^N \frac = \ell+o(1) where :\ell=\sum_^\infty \frac. By summation by parts we have :\begin A(N) &= \sum_^N \frac n^ \\ &= B(N)N^ + \sum_^ B(n) \left (n^-(n+1)^ \right ) \\ &= (B(N)-\ell)N^ + \sum_^ (B(n)-\ell) \left (n^-(n+1)^ \right ) \\ &= o(N^) + \sum_^ \mathcal(n^) \\ &= o(N^) \end :Proposition 2. Define ::L = \begin \sum_^\infty a_n & \text \\ 0 & \text \end :Then: ::\sigma = \limsup_ \frac= \inf_ \left\ :is the abscissa of convergence of the Dirichlet series. ''Proof.'' From the definition :\forall \varepsilon > 0 \qquad A(N)-L = \mathcal(N^) so that :\begin \sum_^N \frac &= A(N) N^ + \sum_^ A(n) (n^ -(n+1)^) \\ &= (A(N)-L) N^ + \sum_^ (A(n)-L) (n^ -(n+1)^) \\ &= \mathcal(N^) + \sum_^ \mathcal(n^) \end which converges as N \to \infty whenever \Re(s) > \sigma. Hence, for every s such that \sum_^\infty a_n n^ diverges, we have \sigma \ge \Re(s), and this finishes the proof. :Proposition 3. If \sum_^\infty a_n converges then f(\sigma+it)= o\left(\tfrac\right) as \sigma \to 0^+ and where it is meromorphic (f(s) has no poles on \Re(s) = 0). ''Proof.'' Note that :n^ - (n+1)^ = sn^+O(n^) and A(N) - f(0) \to 0 we have by summation by parts, for \Re(s) > 0 :\begin f(s) &= \lim_ \sum_^N \frac \\ &= \lim_ A(N) N^ + \sum_^ A(n) (n^-(n+1)^) \\ &= s \sum_^\infty A(n) n^+\underbrace_ \end Now find ''N'' such that for ''n'' > ''N'', , A(n)-f(0), < \varepsilon :s\sum_^\infty A(n) n^ = \underbrace_ + \underbrace_ and hence, for every \varepsilon >0 there is a C such that for \sigma > 0: :, f(\sigma+it), < C+\varepsilon , \sigma+it, \frac.


Formal Dirichlet series

A formal Dirichlet series over a ring ''R'' is associated to a function ''a'' from the positive integers to ''R'' : D(a,s) = \sum_^\infty a(n) n^ \ with addition and multiplication defined by : D(a,s) + D(b,s) = \sum_^\infty (a+b)(n) n^ \ : D(a,s) \cdot D(b,s) = \sum_^\infty (a*b)(n) n^ \ where : (a+b)(n) = a(n)+b(n) \ 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 : (a*b)(n) = \sum_ a(k)b(n/k) \ 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 :F(s) =\sum_^\infty \frac it is possible to show that :F'(s) =-\sum_^\infty \frac 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'') > σ0, then one has that :\frac = - \sum_^\infty \frac converges for Re(''s'') > σ0. 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 : F(s)= \sum_^\infty f(n)n^ and : G(s)= \sum_^\infty g(n)n^. If both ''F''(''s'') and ''G''(''s'') are absolutely convergent for ''s'' > ''a'' and ''s'' > ''b'' then we have : \frac 1 \int_^T \,F(a+it)G(b-it)\,dt= \sum_^\infty f(n)g(n)n^ \textT \sim \infty. If ''a'' = ''b'' and ''ƒ''(''n'') = ''g''(''n'') we have : \frac 1 \int_^T , F(a+it), ^2 \, dt= \sum_^\infty (n)2 n^ \text T \sim \infty.


Coefficient inversion (integral formula)

For all positive integers x \geq 1, the function ''f'' at ''x'', f(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 \sigma > \sigma_, the abscissa of absolute convergence of the DGF ''F'' :f(x) = \lim_ \frac \int_^ x^ F(\sigma + i t) dt. 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 c,x > 0 and any real \Re(s) \equiv \sigma > \sigma_-c where we denote \Re(s) := \sigma: :^ \frac = \frac \int_^ D_f(s+z) \frac dz.


Integral and series transformations

The inverse Mellin transform of a Dirichlet series, divided by s, is given by Perron's formula. Additionally, if F(z) := \sum_ f_n z^n 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 :\sum_ \frac = \frac \int_0^1 \log^(t) F(tz) \, dt,\ s \geq 1. 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 ''an'' generated by a Dirichlet series generating function corresponding to: :\zeta(s)^m = \sum_^\infty \frac where ''ζ''(''s'') is the Riemann zeta function, has the ordinary generating function: :\sum_^\infty a_n x^n = x + \sum_^\infty x^a + \sum_^\infty \sum_^\infty x^ + \sum_^\infty \sum_^\infty \sum_^\infty x^ + \sum_^\infty \sum_^\infty \sum_^\infty \sum_^\infty x^ +\cdots


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 :S_f(x) := \begin \sum_ f(n), & x \geq 1; \\ 0, & 0 < x < 1, \end then we can express ''F'' by the Mellin transform of the summatory function at -s. Namely, we have that :F(s) = s \cdot \int_1^ \frac dx, \Re(s) > \sigma_. For \sigma := \Re(s) > 0 and any natural numbers N \geq 1, we also have the approximation to the DGF ''F'' of ''f'' given by :F(s) = \sum_ f(n) n^ - \frac + s \cdot \int_N^ \frac dy.


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