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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 for any positive integer , 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, .
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 such that , and the corresponding Dirichlet inverse functions, , by inversion formulas involving the summatory function, defined by
In particular, provided that the DGF of some arithmetic function f has an analytic continuation to , 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.
Recall that an arithmetic function is Dirichlet invertible, or has an inverse with respect to Dirichlet convolution such that , or equivalently , if and only if . It is not difficult to prove that is is the DGF of f and is absolutely convergent for all complex s satisfying , then the DGF of the Dirichlet inverse is given by and is also absolutely convergent for all . The positive real 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:
Using the same convention in expressing the result of Perron's formula, we assume that the summatory function of a (Dirichlet invertible) arithmetic function , is defined for all real 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 :
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:
For any s such that , 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 and any real where we denote :
This proof shows that the function can be recovered from its associated Dirichlet series by means of an integral, which is known as the classical integral formula for Dirichlet inversion.
If we are interested in expressing formulas for the Dirichlet inverse of f, denoted by whenever , we write . Then we have by absolute convergence of the DGF for any that
Now we can call on integration by parts to see that if we denote by denotes the antiderivative of F, for any fixed non-negative integers , we have
Thus we obtain that
We also can relate the iterated integrals for the 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
Suppose that we wish to treat the integrand integral formula for Dirichlet coefficient inversion in powers of where , 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 :
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: [1]
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).
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