Multiplication of Dirichlet Series

One of the most beautiful aspects of Dirichlet series is how their multiplication corresponds exactly to Dirichlet convolution of coefficient sequences.

TheoremDirichlet Series Multiplication

Let $F(s) = \sum_{n=1}^{\infty} a_n/n^s$ and $G(s) = \sum_{n=1}^{\infty} b_n/n^s$ be Dirichlet series converging absolutely for $\Re(s) > \sigma_1$ and $\Re(s) > \sigma_2$ respectively.

Then their product converges absolutely for $\Re(s) > \max(\sigma_1, \sigma_2)$ and:

F(s) \cdot G(s) = \sum_{n=1}^{\infty} \frac{c_n}{n^s}

where $c_n = \sum_{d|n} a_d b_{n/d} = (a * b)(n)$ is the Dirichlet convolution.

ExampleZeta Function Identities

The multiplication theorem immediately gives:

$\zeta(s)^2 = \sum_{n=1}^{\infty} \frac{\tau(n)}{n^s}$ (since $\tau = 1 * 1$ )
$\zeta(s) \cdot \frac{1}{\zeta(s)} = 1$ , so $\sum_{n=1}^{\infty} \mu(n)/n^s = 1/\zeta(s)$
$\frac{\zeta(s-1)}{\zeta(s)} = \sum_{n=1}^{\infty} \frac{\varphi(n)}{n^s}$ (since $\varphi * 1 = \text{id}$ )

ExampleGenerating Arithmetic Functions

We can generate new arithmetic functions via convolution:

Let $f * g = h$ . Then $F(s) \cdot G(s) = H(s)$
If $f(n) = 1$ for all $n$ , then $F(s) = \zeta(s)$
If $g(n) = \mu(n)$ , then $G(s) = 1/\zeta(s)$
So $h = 1 * \mu = \delta$ gives $H(s) = 1$ , confirming $\zeta(s) \cdot (1/\zeta(s)) = 1$

Remark

This theorem transforms the ring structure of arithmetic functions (under Dirichlet convolution) into the multiplicative structure of Dirichlet series. It's a form of "transform theory" analogous to how Fourier/Laplace transforms convert convolution to multiplication.

ExampleInverting Dirichlet Series

To find the inverse of $F(s)$ , we need $G(s)$ such that $F(s) \cdot G(s) = 1$ . This means:

(a * b)(n) = \delta(n) = [n=1]

This is solvable recursively:

$b_1 = 1/a_1$
For $n > 1$ : $b_n = -\frac{1}{a_1} \sum_{d|n, d<n} a_{n/d} b_d$

This requires $a_1 \neq 0$ .

The multiplication theorem is fundamental for translating arithmetic identities (convolution relations) into analytic identities (products of Dirichlet series), enabling powerful analytical techniques to study arithmetic functions.