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Dirichlet Generating Function

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

f(s)=sum_(n=1)^(infty)(a_n)/(n^s)
(1)
=a_1+(a_2)/(2^s)+(a_3)/(3^s)+...
(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, zeta(s) gives the sequence of all 1s, while [zeta(s)]^2 gives the sequence of the number of divisors d(n)=sigma_0(n), where sigma_0(n) is the zeroth order divisor function. In the table, mu(n) is the Möbius function, H(n) is the number of ordered factorizations, phi(n) is the totient function, lambda(s) is the Dirichlet lambda function, psi(n) is the Dedekind function, and P(s) is the prime zeta function. In general, [zeta(s)]^k generates the number of ordered factorizations of n into k factors (Wilf 1994, p. 58).

f(s)a_nOEISsequence
1/zeta(s)mu(n)A0086831, -1, -1, 0, -1, 1, -1, 0, 0, 1, ...
zeta(s)11, 1, 1, 1, 1, 1, 1, 1, ...
[zeta(s)]^2d(n)A0000051, 2, 2, 3, 2, 4, 2, 4, ...
zeta(s)zeta(s-k)sigma_k(n)
zeta(s-1)/zeta(s)phi(n)A0000101, 1, 2, 2, 4, 2, 6, 4, 6, 4, 10, ...
1/[2-zeta(s)]H(n)A0020331, 1, 1, 2, 1, 3, 1, 4, 2, 3, 1, 8, ...
lambda(s)1/2[1-(-1)^n]A0000351, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, ...
(zeta(s)zeta(s-1))/(zeta(2s))psi(n)A0016151, 3, 4, 6, 6, 12, 8, 12, 12, 18, 12, ...
(zeta(3s)zeta(s-1))/(zeta(3s-3))cubefree part of nA0509851, 2, 3, 4, 5, 6, 7, 1, 9, 10, 11, 12, ...
P(s)characteristic function of the prime numbers p_nA0000000, 1, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, ...
zeta(s)-1-P(s)characteristic function of the composite numbers c_nA0000000, 0, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, ...

If A(s) and B(s) 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

a_n=sum_(d|n)b_d
(3)

for n>=1. Then

A(s)=B(s)zeta(s),
(4)

and the sequences are related by the Möbius inversion formula

b_n=sum_(d|n)mu(n/d)a_d,
(5)

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

See also

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