In this post I will talk about a few identities, and a mysterious question which was solved a few days ago by Joseph Horan and I. For an arithmetic function , we write
for the Dirichlet series associated to . Observe at once that the Riemann zeta function is given by
where for all . The ubiquity of the zeta function in number-theoretic relationships is due perhaps to this fact — it is the simplest Dirichlet series, just as is one of the simplest arithmetic functions. Similarly, where for all . Also, where is the Mobius function, indeed we have for that
This last fact is a consequence of the behaviour of multiplication of Dirichlet series. Namely, if two Dirichlet series and converge absolutely for some value of , then so too does their product, and indeed . That is, multiplying Dirichlet series amounts to applying Dirichlet convolution to the corresponding coefficients.
Using the behaviour of multiplication of Dirichlet series, we can derive many interesting identities. For example, because the divisor sum function satisfies the relation , and the divisor count function satisfies the relation , we obtain that
and so on. Now, let be the Liouville function, that is, define , where is the number of prime divisors of (counting multiplicity). For example, and . Observe that when we restrict to the square-free integers.
It turns out that we have the relation , where is the Dirichlet identity. Let us say that is -free if no prime appears in the factorization of with a multiplicity of or more. We usually say “square-free” rather than “-free”. Then is nothing more than the indicator function of the square-free integers.
Let denote the indicator function of the -free integers, so that . It is quite clear that is multiplicative. Observe can be written as the Euler product
and therefore, note that
We have therefore demonstrated the relation
As I stated above, the Liouville function , whose definition seems innocently unrelated to the number , is the Dirichlet inverse of . What is going on here? Actually, there is one thing we know, particularly that
Let denote the indicator function of the perfect th powers. Then the above reads (be wary of the important distinction between “perfect th power” and “-free integer”). Notice that has associated Dirichlet series
We want to find, for each , a function such that . For example, we will have , due to the above discussion. However, notice that because we showed above (see ) that , we know that
Hence, suppose that is the Dirichlet inverse of . We know that such a thing exists and must be multiplicative, since is multiplicative, and the multiplicative functions form a subgroup (hence are closed under inversion). Then we must have . Mobius inversion then allows us to write
Let’s assume is a prime power. Note that since vanishes whenever its argument is not squarefree, we can restrict our attentions to the divisors of such that or . A bit of consideration convinces us that the above sum will only ever have one term. Indeed,
Having determined the behaviour of on prime powers, due to its multiplicativity we now understand it fully. The Dirichlet series associated to are