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
.
Related facts
- Product formula for Dirichlet series of completely multiplicative function
- Product formula for Dirichlet series of multiplicative function
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.