Dirichlet Generating Function -- from Wolfram MathWorld

Given a sequence ${a_n}_(n=1)^infty$ , a formal power series

			(1)
			(2)

is called the Dirichlet generating function of the sequence (Wilf 1994, p. 56).

The Dirichlet generating function of a sequence ${a_n}_(n=1)^infty$ can be found in the Wolfram Language using DirichletTransform[a[n], n, s].

The following table summarizes the sequences generated by a number of functions. For example, gives the sequence of all 1s, while gives the sequence of the number of divisors , where is the zeroth order divisor function. In the table, is the Möbius function, is the number of ordered factorizations, is the totient function, is the Dirichlet lambda function, is the Dedekind function, and is the prime zeta function. In general, generates the number of ordered factorizations of into factors (Wilf 1994, p. 58).

	OEIS	sequence
	A008683	1, , , 0, , 1, , 0, 0, 1, ...
1		1, 1, 1, 1, 1, 1, 1, 1, ...
	A000005	1, 2, 2, 3, 2, 4, 2, 4, ...

	A000010	1, 1, 2, 2, 4, 2, 6, 4, 6, 4, 10, ...
	A002033	1, 1, 1, 2, 1, 3, 1, 4, 2, 3, 1, 8, ...
	A000035	1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, ...
	A001615	1, 3, 4, 6, 6, 12, 8, 12, 12, 18, 12, ...
cubefree part of	A050985	1, 2, 3, 4, 5, 6, 7, 1, 9, 10, 11, 12, ...
characteristic function of the prime numbers	A000000	0, 1, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, ...
characteristic function of the composite numbers	A000000	0, 0, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, ...

If and are Dirichlet generating functions of two sequences ${a_n}_(n=1)^infty$ and ${b_n}_(n=1)^infty$ respectively such that the two sequences are connected by