It is a special case of general Dirichlet series.
Dirichlet series play a variety of important roles in analytic number theory.
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.
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
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.
, then the DGF of the inverse function is given by the reciprocal of F: Other identities include where
Examples of Dirichlet series DGFs corresponding to additive (rather than multiplicative) f are given here for the prime omega functions
, which respectively count the number of distinct prime factors of n (with multiplicity or not).
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.
is the Dirichlet inverse of f and where the arithmetic derivative of f is given by the formula
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.
Note that: and define where By summation by parts we have Proof.
:[2] 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 sum and is the Dirichlet convolution of a and b.
The non-zero multiplicative functions form a subgroup of the group of units of Ω.
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 For all positive integers
, the abscissa of absolute convergence of the DGF F [4] 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.
: The inverse Mellin transform of a Dirichlet series, divided by s, is given by Perron's formula.
, then an integral representation for the Dirichlet series of the generating function sequence,
, is given by [5] 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.