Dirichlet series Guide, Meaning , Facts, Information and Description

In mathematics, a Dirichlet series, one of a number of concepts named in honor of Johann Peter Gustav Lejeune Dirichlet, is a series of the form

The most famous of Dirichlet series is

which is the Riemann zeta function.

Other Dirichlet series are:

where μ(n) is the Möbius function;,

where φ(n) is the totient function, and

where σ_a(n) is the divisor function

Analytic properties of Dirichlet series

Given a sequence {a_n}_{n ∈ N} of complex numbers we try to consider the value of

as a function of the complex variable s. In order for this to make sense, we need to consider the convergence properties of the above infinite series:

Theorem. Suppose {a_n}_{n ∈ N} is a bounded sequence of complex numbers. Then the above infinite series for f converges absolutely on the open half-plane of s such that Re(s'') > 1.

If the set of sums a_n+a_n+1+ ... + a_n+k are bounded for n and k ≥ 0, then the above infinite series converges on the open half-plane of s such that such that Re(s) > 0.

In both cases f is an analytic function on the corresponding open half plane.

In many cases, the analytic function associated with a dirichlet series has an analytic extension to a larger domain. This is the case for the zeta function:

Theorem. The zeta function has a meromorphic extension to C with a unique pole at s=1.

One of the most important (unsolved) conjectures of mathematics called the Riemann hypothesis concerns the zeros of the zeta function.

See also

• Dirichlet

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