## Monday Math 94

Consider two arithmetic functions f(n) and g(n), with Dirichlet series generating functions F(s) and G(s), respectively. What, then, can we say about the product F(s)G(s)?

Using different summation indicies a and b, we have
 and . The product, then, is
.

Now, suppose we reorder the sum so as to group terms with the same value of ab. We let n=ab; then we have , and there is one term with given n for each a which divides that n, so within each set of terms with the same n, we sum over a|n, and we get:,
and so we see the product F(s)G(s) is the Dirichlet series generating function of the Dirichlet convolution of f(n) and g(n) (compare to the relationship between the convolution and the Fourier transform).

Now, this implies that the Dirichlet series generating function of the unit function ε(n) is 1, which should be obvious, as
.
And since , we see that if f(n) has Dirichlet series generating function F(s), then its Dirichlet inverse f-1(n) has Dirichlet series generating function 1/F(s). For example, consider the Dirichlet inverses 1(n) and μ(n), which have Dirichlet series generating functions  and , respectively. You may also recall that here I showed that the Dirichlet series generating function for |μ| is . I also demonstrated here that |μ| and the Liouville function λ(n) are Dirichlet inverses, so we can see that λ(n) has Dirichlet series generating function ; we can confirm this via the Euler product for Dirichlet series generating functions of completely multiplicative functions.

One should also note that if a function f(n) has Dirichlet series generating function F(s), then the function  has Dirichlet series generating function F(sa); since
,
then
.
For example, the Dirichlet series generating function of  is thus .

Combining this with our above relationship between Dirichlet convolution and Dirichlet series generating functions, we can develop a number of proofs. For example, the proof that the Dirichlet series generating function of the divisor function σx(n) is  becomes much more simple than the proof seen here:
Since , we see ; since the Dirichlet series generating function of 1(n) is , and the Dirichlet series generating function of  is , the Dirichlet series generating function of σx(n) is their product, .
Similarly, we can use the Möbius inversion formula relationship that , found here, to confirm that φ(n) has Dirichlet series generating function , as we found via Euler product here.