Dirichlet series are functions of a complex variable that are defined by certain infinite series. They are generalizations of the Riemann zeta function, and are important in number theory due to their deep connections with the distribution of prime numbers. They have interesting connections with multiplicative functions and Dirichlet convolution.
A Dirichlet series is an expression of the form
Here is a complex variable and is a sequence of complex numbers. If we write the as the values of an arithmetic function we say that the above series is the Dirichlet series associated with
When for all the associated Dirichlet series is the Riemann zeta function So the Dirichlet series associated with the function is
Given two Dirichlet series and , the Dirichlet series representation of the product turns out to be related to Dirichlet convolution.
Suppose that the series both converge for a given value of and at least one of them converges absolutely. Then converges, where is the Dirichlet convolution of and , and
Ignoring questions of convergence, consider the product
This expands to give a sum of fractions of the form
The terms with denominator will be ones where So the coefficient of will be which is just .
We can generalize this to any Dirichlet series associated with a multiplicative function:
Given that converges absolutely,
- if is a multiplicative function, then
- if is completely multiplicative, then
The proof is essentially the same as the above derivation for the Riemann zeta function.
The product formula gives
which is the reciprocal of the product formula for . Another way to solve the problem is to use the result on products of Dirichlet series: note that the convolution identity function, so the product of the associated Dirichlet series is .
where is the Euler phi function.
, where (see Dirichlet convolution for details), so the theorem above gives
(Exercise: do this example a different way, by using the product formula as in the previous example.)