# Dirichlet Generating Functions

Franklin T. Adams-Watters franktaw at netscape.net
Wed Sep 21 06:16:19 CEST 2005

```I wrote:
>A034448 (Sum of unitary divisors)
>    zeta(s)*zeta(s-1)/zeta(2s-1)
>    [Somebody should check this; I'm not 100% confident of my
>    derivation.]
>    Should note that this is multiplicative with
>    a(p^e) = p^e+1 for e>0.

I have now verified this formula.

To verify the Dirichlet generating function for a multiplicative function, you can define (in PARI):

z(a, b) = 1/(1-x^a*p^-b)+O(x^16)

Then take your candidate formula (which must be a product of terms like zeta(a*s+b)^k), and replace all the zeta(a*s+b) with z(a,b).  The result should be the function with a(p^n) as the coefficient of x^n.  In this case, we get

z(1,0)*z(1,-1)/z(2,-1)

which PARI evaluates to:

1 + (p + 1)*x + (p^2 + 1)*x^2 + (p^3 + 1)*x^3 + (p^4 + 1)*x^4 + (p^5 + 1)*x^5 + (p^6 + 1)*x^6 + (p^7 + 1)*x^7 + (p^8 + 1)*x^8 + (p^9 + 1)*x^9 + (p^10 + 1)*x^10 + (p^11 + 1)*x^11 + (p^12 + 1)*x^12 + (p^13 + 1)*x^13 + (p^14 + 1)*x^14 + (p^15 + 1)*x^15 + O(x^16)

Which matches the formula given.

I can also add one more Dirichlet G.F.:
A112526: (Characteristic function of powerful numbers)
zeta(2*s)*zeta(3*s)/zeta(6*s)
(I just added this sequence 11 days ago, but I didn't know the D.G.F. at the time.)
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