Dirichlet series
In mathematics, a Dirichlet series is any series of the form formula_1 where "s" is complex, and formula_2 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. 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 "a" 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 formula_7 Examples. The most famous example of a Dirichlet series is whose analytic continuation to formula_9 (apart from a simple pole  at formula_10) 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 formula_11: 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. 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 where is a Dirichlet L-function. If the arithmetic function has a Dirichlet inverse function formula_16, i.e., if there exists an inverse function such that the Dirichlet convolution of "f" with its inverse yields the multiplicative identity formula_17, then the DGF of the inverse function is given by the reciprocal of "F": Other identities include where formula_20 is the totient function, where "J" is the Jordan function, and where "σ"("n") is the divisor function. By specialization to the divisor function "d" = "σ  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 formula_32 and formula_33, 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 formula_34: If "f" is a multiplicative function such that its DGF "F" converges absolutely for all formula_36, and if "p" is any prime number, we have that where formula_38 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 formula_40, then by Moebius inversion we have that formula_41. 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 formula_43 is the DGF of some arithmetic f" with formula_44, then the DGF "G" is expressed by the sum where formula_16 is the Dirichlet inverse of "f" and where the arithmetic derivative of "f" is given by the formula formula_47 for all natural numbers formula_48. Analytic properties. Given a sequence formula_49 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 formula_49 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 "a" = O("n"), 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 formula_53 is the abscissa of convergence of a Dirichlet series if it converges for formula_54 and diverges for formula_55 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 converges for some formula_57 "Proof." Note that: and define where By summation by parts we have "Proof." From the definition so that  which converges as formula_67 whenever formula_68 Hence, for every formula_69 such that formula_70 diverges, we have formula_71 and this finishes the proof. "Proof." Note that and formula_78 we have by summation by parts, for formula_79 Now find "N" such that for "n" > "N", formula_81 and hence, for every formula_83 there is a formula_84 such that for formula_85: 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 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. 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, then one has that converges for Re("s") > σ. Here, Λ("n") is the von Mangoldt function. 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 Coefficient inversion (integral formula). For all positive integers formula_99, the function "f" at "x", formula_100, can be recovered from  the DGF "F" of "f" (or the Dirichlet series over "f") using the following integral formula whenever formula_101, 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 formula_103 and any real formula_104 where we denote formula_105: Integral and series transformations. The inverse Mellin transform of a Dirichlet series, divided by s, is given by Perron's formula. Additionally, if formula_107 is the (formal) ordinary generating function of the sequence of formula_108, then an integral  representation for the Dirichlet series of the generating function sequence, formula_109, 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" generated by a Dirichlet series generating function corresponding to: where "ζ"("s") is the Riemann zeta function, 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 of the summatory function at formula_114. Namely, we have that For formula_116 and any natural numbers formula_117, we also have the approximation to the DGF "F" of "f" given by