Dirichlet series
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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
- <math>f(s)=\sum_{n=1}^{\infty} \frac{a_n}{n^s}.<math>
The most famous of Dirichlet series is
- <math>\zeta(s)=\sum_{n=1}^{\infty} \frac{1}{n^s},<math>
which is the Riemann zeta function.
Other Dirichlet series are:
- <math>\frac{1}{\zeta(s)}=\sum_{n=1}^{\infty} \frac{\mu(n)}{n^s}<math>
where μ(n) is the Möbius function,
- <math>\frac{\zeta(s-1)}{\zeta(s)}=\sum_{n=1}^{\infty}
\frac{\varphi(n)}{n^s}<math>
where φ(n) is the totient function, and
- <math>\zeta(s) \zeta(s-a)=\sum_{n=1}^{\infty} \frac{\sigma_{a}(n)}{n^s}<math>
- <math>\frac{\zeta(s)\zeta(s-a)\zeta(s-b)\zeta(s-a-b)}{\zeta(2s-a-b)}
=\sum_{n=1}^{\infty} \frac{\sigma_a(n)\sigma_b(n)}{n^s}<math>
where σa(n) is the divisor function
Analytic properties of Dirichlet series
Given a sequence {an}n ∈ N of complex numbers we try to consider the value of
- <math> f(s) = \sum_{n=1}^\infty \frac{a_n}{n^s} <math>
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 {an}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 an + an + 1 + ... + an + k is 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 general the abscissa of convergence of a Dirichlet series is the intercept on the real axis of the vertical line in the complex line, such that there is convergence to the right of it, and divergence to the left. This is the analogue for Dirichlet series of the radius of convergence for power series. The Dirichlet series case is more complicated, though: absolute convergence and uniform convergence may occur in distinct half-planes.
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 open conjectures of mathematics — the Riemann hypothesis — concerns the zeroes of the zeta function.