prime sig and multiplicative

[Christian G.Bower]

Mitch Harris wrote:

> - Some comments mention that the sequence is dependent only on the "prime
> signature" (or something similar) which means that it is multiplicative.

No, prime signature does not mean multiplicative, it means any
sequence with the same powers on (possibly different) primes
has the same value as in a(2^2*3^3) = a(7^2*5^3)

A001055 factorizations of n is prime signature but not multiplicative.

I have a few more musings about prime sigs and mult.

I imagine much of this is in some number theory book I'll never read,
but I just discovered it and intend on having some fun with it.

This whole business of looking at Dirichlet series for sequence and
treating numbers by their factors is like taking a vector space
of integers of unbounded dimension. (Okay, I guess it's really a
module since integers don't form a field, but that's not relevant
here.)  So I thought I'd look at the 2-d case for simplicity.

If I have a 2-d sequence such that a(i,j)=a(i,0)*a(0,j) then the
ordinary generating function satisfies A(x,y)=B(x)*C(y) for some
B and C, hence it can be decomposed into a product of 1-d terms.

Translating to the Dirichlet series of a multiplicative sequence,
I have A(x) = A_2(x)*A_3(x)*A_5(x)*...
for components A_2(x)... that each only have coefficients in the
p^k terms for their prime number.

For a prime signature sequence, in the 2-d world it corresponds
to symmetric (or is that commutative) tables: a(i,j)=a(j,i) and
A(x,y)=A(y,x).

Those that are both prime signature and multiplicative would be
like A(x,y)=B(x)*B(y) for some B.


Another thing I've been looking at is product versions of
sequence transforms.  Setprod, from my last posting is just the
product version of the Euler transform.  There are quite a few
transforms that I would describe as "combinatorial compositions."
Most of these can be described by an index series which is a
formal polynomial series of the variables x_1, x_2, x_3,...

If S(x_1,x_2,x_3,...) is the index series of a transform, then
the transform affects the ordinary generating function as follows:
T(A(x)) = S(A(x), A(x^2), A(x^3),...)

Hence, Euler transform with index series:

exp(x_1/1 + x_2/2 + x_3/3 + ...)

becomes
Euler(A(x)) = exp(A(x)/1 + A(x^2)/2 + A(x^3)/3 + ...)

Invert transform with index series 1/(1 - x_1) becomes
Invert(A(x)) = 1/(1-A(x))

The index series can also be used to define the product transforms
to the Dirichlet generating function.

T(A(s)) = S(A(s), A(2s), A(3s),...)

thus we have

setprod(A(s)) = exp(A(s)/1 + A(2s)/2 + A(3s)/3 + ...)

lineprod(A(s)) = 1/(1-A(s))

and this process can be performed on any transform described by
an index series (like the Moebius transforms, the cycle transforms,
the Weigh transform, the Carlitz transform ...)

Up till now, I've only created product sequences using those two
product transforms and the product version of the Weigh transform.
I don't think there is a whole lot of study going on in how one
might arrange factors into different structures, but maybe there
should be.  Another problem with factorization sequences is they
often have to go a large number of terms (like 60 or more) before
they are distinguished from similar sequences.

Enough rambling for now.

Christian