Mathematica's Help documentation on
DirichletConvolve is economical, to say the least. It claims the function "gives the Dirichlet convolution of the expressions f and g" and that's it, except for the example
In:= DirichletConvolve[n, n, n, m]
Out= m DivisorSigma[0, m]
This is going to sound stupid, but here's what I take away from the help documentation:
According to their own definition, the expression
DirichletConvolve[n, n, n, m] convolves a function
n with some other function also called
n to produce an output which ignores the third
n completely, and produces a specific function of
m as output - even though the original two separate functions both defined by
n have not been specified...
So, obviously I have it wrong. But I genuinely cannot make any sense of it. Specifically:
- Dirichlet convolution convolves two functions
g[n](with the same argument
n) to produce
Sum[f[d]*g[n/d], Divisible[n, d]]. Where does a fourth input quantity
mfeature in this? And where does the third quantity
ndisappear to? In some expressions, changing the third quantity does make a difference, but I can't riddle what that difference is, and why. This is because...
- In every case, the output seems to be a sum of multiples of
DivisorSigma[0, m](with some of those multiples being factors of
nderived through some equality that I'm unaware of. This is emphatically not the case with actual Dirichlet convolution, so what gives?
I'm sure that these are dumb questions and there's a simple explanation, but it isn't there in the Help documentation.
Can anyone explain, perhaps with a worked-through example? It's a big ask, I know, but I'd really appreciate it.