Dirichlet Generating Functions

[Franklin T. Adams-Watters]

There is a serious shortage of Dirichlet generating functions in the
OEIS.  Some sequences that should have generating functions but don't
include:

A000012 (Ones sequence)
    zeta(s)
A000027 and A001477 (Natural numbers)
    zeta(s-1)
    A001477 is also missing the E.G.F. x*e^x.
A000578 (Cubes)
    zeta(s-3)
    Also missing E.G.F. (x+3x^2+x^3)*e^x
A000583 (Fourth powers)
    zeta(s-4)
    The E.G.F. shown for this sequence is apparently belongs
    with some other sequence, since it is a 2D G.F.
    The E.G.F. for this sequence is (x+7x^2+6x^3+x^4)*e^x.
    [This sequence shows the general form for the O.G.F. of n^m.
    It should probably also show the general form for the E.G.F,
    which is phi_m(x)*e^x, where phi_m is the exponential
    polynomial of order n.]
Higher powers in A001014 to A001017, A008454 to A008456, and A010801
to A010813 are missing all three generating functions.
A010052 (Characteristic function of squares)
    zeta(2s)
A037213 (Square root if square, else zero)
    zeta(2s-1)
A000010 (Totient or phi function)
    zeta(s-1)/zeta(s)
A000005 (Divisor or tau function)
    zeta(s)^2
A008683 (Mobius mu function)
    [This has the Dirichlet G.F., but not identified as such.]
    [Also should add Dirichlet G.F. for the absolute value:]
    zeta(s)/zeta(2s)
A010057 (Characteristic function of cubes)
    zeta(3s)
A017665 (Numerators of sigma_{-1}(n))
    zeta(s)*zeta(s+1)
    [For fraction A017665/A017666]
A017667 (Numerators of sigma_{-2}(n))
    zeta(s)*zeta(s+2)
    [For fraction A017667/A017668]
...
A017711 (Numerators of sigma_{-24}(n))
    zeta(s)*zeta(s+24)
    [For fraction A017711/A017712]
A007434 (Mobius transform of squares)
    zeta(s-2)/zeta(s)
A059376 (Mobius transform of cubes)
    zeta(s-3)/zeta(s)
A059377 (Mobius transform of 4th powers)
    zeta(s-4)/zeta(s)
A059378 (Mobius transform of 5th powers)
    zeta(s-5)/zeta(s)
A034444 (Number of unitary divisors)
    zeta(s)^2/zeta(2s)
    [Should also note that this is the inverse Mobius transform
    of A008966]
A055615 (n times mu(n))
    [This has the Dirichlet G.F.  It should also have the
    Dirichlet G.F. for the absolute value:]
    zeta(s-1)/zeta(2s-2)
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.

The next few reference the Prime Zeta function, sometimes called
P(s), but which I will specify as primezeta(s).
Cf http://mathworld.wolfram.com/PrimeZetaFunction.html.
This reference should probably be added for these sequences.

A010051 (Characteristic function of primes)
    primezeta(s)
A001221 (Number of prime divisors; omega(n))
    zeta(s)*primezeta(s)
A061397 (n if prime, else zero)
    primezeta(s-1)

The next few sequences use a function which I have not seen described
anywhere (which doesn't mean that nobody has looked at it).  This is
naturally described as the Prime Power Zeta function, and I will
designate it as ppzeta(s).  For purposes of this function, we do not
consider 1 a prime power; in other words, we are looking at numbers
p^k where k>0.  We then have:

    ppzeta(s) = sum_{p prime} sum_{k=1}^{infinity} 1/(p^)k^s

Summing this as written, we get:

    ppzeta(s) = sum_{p prime} 1/(p^s-1)

Reversing the order of summation, we get:

    ppzeta(s) = sum_{k=1}^{infinity} primezeta(k*s)

That's as far as I have been able to get in analyzing this function.

A069513 (Characteristic function for prime powers excluding one)
    ppzeta(s)
    [I think the name for this sequence should be the description
    I used here or something similar; the current name should be
    moved to the Formulas area, also noting that this is the
    Mobius inversion formula.]
    [This should reference A001222 (big omega) in the See also
    section.]
A010055 (Characteristic function for prime powers including one)
    1+ppzeta(s)
A001222 (Number of prime divisors w/ Multiplicity - big omega)
    ppzeta(s)*zeta(s)

-- 
Franklin T. Adams-Watters
16 W. Michigan Ave.
Palatine, IL 60067
847-776-7645


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