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 d0 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:

Read more about this topic:  Dirichlet Series

Famous quotes containing the word examples:

    It is hardly to be believed how spiritual reflections when mixed with a little physics can hold people’s attention and give them a livelier idea of God than do the often ill-applied examples of his wrath.
    —G.C. (Georg Christoph)

    No rules exist, and examples are simply life-savers answering the appeals of rules making vain attempts to exist.
    André Breton (1896–1966)

    There are many examples of women that have excelled in learning, and even in war, but this is no reason we should bring ‘em all up to Latin and Greek or else military discipline, instead of needle-work and housewifry.
    Bernard Mandeville (1670–1733)