i've been marinating the idea of generalizing dirichlet convolutions in several ways and what has been interesting was finding which methods of generalization just boil down to being dirichlet convolutions again under the hood and which don't.
for example, try generalizing by selecting only divisors of a certain form. let h be some function h : N_{>0} -> N_{>0}.
now write the convolution as:
(f * g)(n) = Σ_{ h(d) | n } f(h(d))•g(n / h(d))
if we define an indicator function t(x) to return 1 when x is in the image of h, and 0 otherwise, we can rewrite this as:
(f * g)(n) = Σ_{ d | n ; t(d) = 1 } f(d)•g(n/d)
-> (f * g)(n) = Σ_{ d | n } t(d)•f(d)•g(n/d)
-> (tf * g)(n)
which is just dirichlet convolution again. (this is pretty obvious if u think about it for a minute, but i was exhausted when first working on this & proved it via dirichlet series rather than the simple route lmao)
as far as i've seen so far, the way to generalize to expand upon dirichlet convolutions comes down to information. u need to make it so that the inner functions f,g don't "know" as much about n when passed arguments in the sum. it's a little hard to explain this but it makes sense to me intuitively.
one example of a generalization that's more interesting to me recently is iterating over numbers coprime to n rather than divisors. where a divisor set might look like {1, 3, 5, 15} and pair up as (1,15), (3,5), the set of integers coprime to 15 in this case would be {1, 2, 4, 7, 8, 11, 13, 14} and pair up as (1,14), (2,13), (4,11), (7,8).
so in this case the dirichlet convolution would sum up f(1)g(15) + f(3)g(5) + f(5)g(3) + f(15)g(1), but the generalization would sum up:
f(1)g(14) + f(2)g(13) + f(4)g(11) + f(7)g(8) + f(8)g(7) + f(11)g(4) + f(13)g(2) + f(14)g(1)
which has a different relationship to knowledge of n than dirichlet convolutions do. which is interesting to explore. where dirichlet convolutions pair up (x,y) such that xy = n, this approach pairs up numbers such that x+y = n, which might be interesting for looking for connections between addition & multiplication. additionally, i don't think that this convolution over coprime numbers is able to be represented as a dirichlet convolution, and might deal with a unique space of arithmetic functions.
