In the field of mathematical analysis, a general Dirichlet series is an infinite series that takes the form of where
is a strictly increasing sequence of nonnegative real numbers that tends to infinity.
A simple observation shows that an 'ordinary' Dirichlet series is obtained by substituting
while a power series is obtained when
There are now three possibilities regarding the convergence of a Dirichlet series, i.e. it may converge for all, for none or for some values of s. In the latter case, there exist a
if the series converges everywhere on the complex plane.
The abscissa of convergence of a Dirichlet series can be defined as
Another equivalent definition is The line
The half-plane of convergence is defined as The abscissa, line and half-plane of convergence of a Dirichlet series are analogous to radius, boundary and disk of convergence of a power series.
On the line of convergence, the question of convergence remains open as in the case of power series.
The proof is implicit in the definition of abscissa of convergence.
(alternating harmonic series) and diverges at
Suppose that a Dirichlet series does not converge at
On the other hand, if a Dirichlet series converges at
which can be determined by various convergence tests.
These formulas are similar to the Cauchy–Hadamard theorem for the radius of convergence of a power series.
As usual, an absolutely convergent Dirichlet series is convergent, but the converse is not always true.
If a Dirichlet series is absolutely convergent at
A Dirichlet series may converge absolutely for all, for no or for some values of s. In the latter case, there exist a
The abscissa of absolute convergence can be defined as
above, or equivalently as The line and half-plane of absolute convergence can be defined similarly.
is given by In general, the abscissa of convergence does not coincide with abscissa of absolute convergence.
The width of this strip is given by In the case where L = 0, then All the formulas provided so far still hold true for 'ordinary' Dirichlet series by substituting
It is possible to consider other abscissas of convergence for a Dirichlet series.
, and a remarkable theorem of Bohr in fact shows that for any ordinary Dirichlet series where
[1] Bohnenblust and Hille subsequently showed that for every number
[2] A formula for the abscissa of uniform convergence
[3] A function represented by a Dirichlet series is analytic on the half-plane of convergence.
A Dirichlet series can be further generalized to the multi-variable case where
, k = 2, 3, 4,..., or complex variable case where