Dirichlet series inversion
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. For  now, we will use this page as a compendia of "oddities" and oft-forgotten facts about transforming and inverting Dirichlet series, DGFs, and relating the inversion of a DGF of a sequence to the sequence's summatory function. We also use the notation for coefficient extraction usually applied to formal generating functions in some complex variable, by denoting formula_1 for any positive integer formula_2, whenever denotes the DGF (or Dirichlet series) of "f" which is taken to be absolutely convergent whenever the real part of "s" is greater than the abscissa of absolute convergence, formula_4. The relation of the Mellin  transformation of the summatory function of a sequence to the DGF of a sequence provides us with a way of expressing arithmetic functions formula_5 such that formula_6, and the corresponding Dirichlet inverse functions, formula_7, by inversion formulas involving the summatory function, defined by In particular, provided that the DGF of some arithmetic function "f" has an to formula_9, we can express the Mellin transform of the summatory function of "f" by the continued DGF formula as It is often also convenient to express formulas for the summatory functions over the Dirichlet inverse function of "f" using this construction of a  Mellin inversion type problem. Preliminaries: Notation, conventions and known results on DGFs. DGFs for Dirichlet inverse functions. Recall that an arithmetic function is Dirichlet invertible, or has an inverse formula_7 with respect to Dirichlet convolution such that formula_12, or equivalently formula_13, if and only if formula_6. It is not difficult to prove that is formula_15 is the DGF of "f" and is absolutely convergent for all complex "s" satisfying formula_16, then the DGF of the Dirichlet inverse is given by formula_17 and is also absolutely convergent for all formula_16. The positive real formula_19 associated with each invertible arithmetic function "f  is called the abscissa of convergence. We also see the following identities related to the Dirichlet inverse of some function g" that does not vanish at one: Summatory functions. Using the same convention in expressing the result of Perron's formula, we assume that the summatory function of a (Dirichlet invertible) arithmetic function formula_21, is defined for all real formula_22 according to the formula We know the following relation between the Mellin transform of the summatory function of "f" and the DGF of "f" whenever formula_16: Some examples of this relation include the following identities involving the Mertens function, or  summatory function of the Moebius function, the prime zeta function and the prime-counting function, and the Riemann prime-counting function: Statements of the integral formula for Dirichlet inversion. Classical integral formula. For any "s" such that formula_27, we have that If we write the DGF of "f" according to the Mellin transform formula of the summatory function of "f", then the stated integral formula simply corresponds to a special case of Perron's formula. 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_29  and any real formula_30 where we denote formula_31: Special cases of the formula. If we are interested in expressing formulas for the Dirichlet inverse of "f", denoted by formula_7 whenever formula_6, we write formula_35. Then we have by absolute convergence of the DGF for any formula_16 that Now we can call on integration by parts to see that if we denote by formula_38 denotes the formula_39 antiderivative of "F", for any fixed non-negative integers formula_40, we have Thus we obtain that We also can relate the iterated integrals for the formula_43 antiderivatives of "F" by a finite sum of  k single integrals of power-scaled versions of "F": In light of this expansion, we can then write the partially limiting "T"-truncated Dirichlet series inversion integrals at hand in the form of A formal generating-function-like convolution lemma. Suppose that we wish to treat the integrand integral formula for Dirichlet coefficient inversion in powers of formula_46 where formula_47, and then proceed as if we were evaluating a traditional integral on the real line. Then we have that We require the result given by the following formula, which is proved rigorously by an application of integration by parts, for any  non-negative integer formula_49: So the respective real and imaginary parts of our arithmetic function coefficients "f" at positive integers "x" satisfy: The last identities suggest an application of the Hadamard product formula for generating functions. In particular, we can work out the following identities which express the real and imaginary parts of our function "f" at "x" in the following forms: Notice that in the special case where the arithmetic function "f" is strictly real-valued, we expect that the inner terms in the previous limit formula are always zero (i.e., for any "T").