# Dirichlet series of Dirichlet product equals product of Dirichlet series

## Statement

### For formal Dirichlet series

Suppose  and  are arithmetic functions. Then, the Dirichlet series of the Dirichlet product  equals the product of the Dirichlet series for  and for .

In symbols:

.

### For functions

This identity holds for the functions corresponding to the Dirichlet series whenever these series are absolutely convergent, because the additive rearrangement needed is valid when the series are absolutely convergent. Note that this also shows that if the Dirichlet series for  and  are absolutely convergent for some , the Dirichlet series for  is also absolutely convergent for the same .

Finally, if we consider the analytic continuations of these Dirichlet series to the complex numbers, then the analytic continuation for the function corresponding to  is the product of the analytic continuations for  and for .

## Proof

We expand the right side:

.

Note that every ordered pair  occurs for exactly one , namely . Thus, the above sum can be rearranged as:

.

Relabeling  and , we get the identity we need to prove.