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[1]:= DirichletConvolve[n, n, n, m]

Out[1]= 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 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:

1. Dirichlet convolution convolves two functions f[n] and g[n] (with the same argument n) to produce Sum[f[d]*g[n/d], Divisible[n, d]]. Where does a fourth input quantity m feature in this? And where does the third quantity n disappear 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...
2. In every case, the output seems to be a sum of multiples of DivisorSigma[0, m] (with some of those multiples being factors of n derived 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.

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[1]:= DirichletConvolve[n, n, n, m]

Out[1]= 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:

1. Dirichlet convolution convolves two functions f[n] and g[n] (with the same argument n) to produce Sum[f[d]*g[n/d], Divisible[n, d]]. Where does a fourth input quantity m feature in this? And where does the third quantity n disappear 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...
2. In every case, the output seems to be a sum of multiples of DivisorSigma[0, m] (with some of those multiples being factors of n derived 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.