Dirichlet Series - Examples

Examples

The most famous of Dirichlet series is

which is the Riemann zeta function.

Treating these as formal Dirichlet series for the time being in order to be able to ignore matters of convergence, note that we have:

\zeta(s) = \mathfrak{D}^{\mathbb{N}}_{\mathrm{id}}(s) = \prod_{p\,\mathrm{prime}} \mathfrak{D}^{\{p^n : n \in \mathbb{N}\}}_{\mathrm{id}}(s) = \prod_{p\,\mathrm{prime}} \sum_{n \in \mathbb{N}} \mathfrak{D}^{\{p^n\}}_{\mathrm{id}}(s) = \prod_{p\,\mathrm{prime}} \sum_{n \in \mathbb{N}} \frac{1}{(p^n)^s} = \prod_{p\,\mathrm{prime}} \sum_{n \in \mathbb{N}} \left(\frac{1}{p^s}\right)^n = \prod_{p\,\mathrm{prime}} \frac{1}{1-\frac{1}{p^s}},

as each natural number has a unique multiplicative decomposition into powers of primes. It is this bit of combinatorics which inspires the Euler product formula.

Another is:

where μ(n) is the Möbius function. This and many of the following series may be obtained by applying Möbius inversion and Dirichlet convolution to known series. For example, given a Dirichlet character χ(n) one has

where L(χ, s) is a Dirichlet L-function.

Other identities include

where φ(n) is the totient function,

where Jk is the Jordan function, and

where σa(n) is the divisor function. By specialisation to the divisor function d=σ0 we have

The logarithm of the zeta function is given by

for Re(s) > 1. Here, Λ(n) is the von Mangoldt function. The logarithmic derivative is then

These last two are special cases of a more general relationship for derivatives of Dirichlet series, given below.

Given the Liouville function λ(n), one has

Yet another example involves Ramanujan's sum:

Another example involves the Mobius function:

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