In this note, I consider an application of generalized Mobius Inversion to extract information of arithmetical sums with asymptotics of the form for a fixed and a constant , so that the sum is over both and . We will see that .
For completeness, let’s prove the Mobius Inversion formula. Suppose we have an arithmetic function that has Dirichlet inverse , so that , where I use to denote the indicator function of the condition
Then if and are complex-valued functions on , then
if and only if
Suppose that . Then .
Let’s collect terms. For any given , the number of times will occur will be one for every factorization of (that is, one time for every way of writing ). So reorganizing the sum, we get that it’s equal to . Since we know that and are Dirichlet inverses, the only term that survives is when . So we have, after the dust settles, that our sum is nothing more than . And the converse is the exact same argument. .
We know some Dirichlet inverses already. If is the function sending everything to , i.e. , then we know that , where is the Mobius function. This means that . This also means that we know that .
Let me take a brief aside to mention a particular notation I endorse: the Iverson bracket notation, which says that . Why? Well, a while ago, I really liked to use the bolded 1 indicator function, but then I started doing things with modules and representations, and someone bolded 1 became a multiplicative identity. This is done in Artin’s algebra text too, which I like (but did not learn out of). Then I really liked the -style indicator notation, until it became the default letter for group characters. So I’ve turned to the Iverson Bracket, which saves space compared to the other two anyway.
Back to Dirichlet inverses. Some are a bit more challenging to find. We might say, let’s find the Dirichlet inverse of the function for a fixed , i.e. the th-power indicator function. This is one of those things that we’ll be using later, and I’m acting like one of those teachers who just happens to consider the right thing at the right time.
Claim: the Dirichlet inverse of the function is the function , which through slightly abusive notation is to say the function that is zero on non-th-powers, and takes the value of when .
Possible Motivation of claim: This doesn’t need to have descended as a gift from the gods. Examining on powers makes it seem like we want $\latex \mu$-like function, and a little computation would give a good suggestion as to what to try.
Both sides are on . That’s a good start. Note also that both sides are zero on non--th-powers, so we only consider th powers. For some , consider . Then as . So we are done.
That’s sort of nice. Let’s go on to the main bit of the day.
Proof of lemma:
Proposition: , where .
Proof: We now know that , which is of the form . So by Mobius inversion, .
Let’s look at this last sum. .
For the main term, recall that we know that , so that (asymptotically) we know that .
For the error term, note that , so , and where the convergence of this list sum is guaranteed by the condition on being not too big (in a sense, we can’t maintain arbitrarily small error).
Every step is reversible, so putting it all together we get , as we’d wanted.
In general, if one has error term for some , then this process will yield an error term with , but it will still be true that . I have much more to say about applying Mobius Inversion to asymptotics of this form, and will follow this up in another note.