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# Dirichlet Generating Function

Given a sequence , 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 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 and respectively such that the two sequences are connected by

 (3)

for . Then

 (4)

and the sequences are related by the Möbius inversion formula

 (5)

where is the Möbius function (Wilf 1994, p. 62).

Dirichlet L-Series, Dirichlet Series, Generating Function, Möbius Transform

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## References

Sloane, N. J. A. Sequences A000005/M0246, A000010/M0299, A000035/M0001, A002033/M0131, A008683, and A050985 in "The On-Line Encyclopedia of Integer Sequences."Wilf, H. Generatingfunctionology, 2nd ed. New York: Academic Press, 1994.

## Cite this as:

Weisstein, Eric W. "Dirichlet Generating Function." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/DirichletGeneratingFunction.html