In the field of
mathematical analysis
Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, infinite sequences, series (m ...
, a general Dirichlet series is an
infinite series
In mathematics, a series is, roughly speaking, a description of the operation of adding infinitely many quantities, one after the other, to a given starting quantity. The study of series is a major part of calculus and its generalization, math ...
that takes the form of
:
where
,
are
complex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...
s and
is a strictly increasing
sequence
In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is calle ...
of nonnegative
real numbers
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...
that tends to infinity.
A simple observation shows that an 'ordinary'
Dirichlet series
In mathematics, a Dirichlet series is any series of the form
\sum_^\infty \frac,
where ''s'' is complex, and a_n is a complex sequence. It is a special case of general Dirichlet series.
Dirichlet series play a variety of important roles in analyti ...
:
is obtained by substituting
while a
power series
In mathematics, a power series (in one variable) is an infinite series of the form
\sum_^\infty a_n \left(x - c\right)^n = a_0 + a_1 (x - c) + a_2 (x - c)^2 + \dots
where ''an'' represents the coefficient of the ''n''th term and ''c'' is a const ...
:
is obtained when
.
Fundamental theorems
If a Dirichlet series is convergent at
, then it is
uniformly convergent
In the mathematical field of analysis, uniform convergence is a mode of convergence of functions stronger than pointwise convergence. A sequence of functions (f_n) converges uniformly to a limiting function f on a set E if, given any arbitrarily s ...
in the
domain
Domain may refer to:
Mathematics
*Domain of a function, the set of input values for which the (total) function is defined
**Domain of definition of a partial function
**Natural domain of a partial function
**Domain of holomorphy of a function
* Do ...
:
and
convergent for any
where
.
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
such that the series is convergent for
and
divergent for
. By convention,
if the series converges nowhere and
if the series converges everywhere on the
complex plane
In mathematics, the complex plane is the plane formed by the complex numbers, with a Cartesian coordinate system such that the -axis, called the real axis, is formed by the real numbers, and the -axis, called the imaginary axis, is formed by the ...
.
Abscissa of convergence
The abscissa of convergence of a Dirichlet series can be defined as
above. Another equivalent definition is
:
The line
is called the line of convergence. The half-plane of convergence is defined as
:
The
abscissa
In common usage, the abscissa refers to the (''x'') coordinate and the ordinate refers to the (''y'') coordinate of a standard two-dimensional graph.
The distance of a point from the y-axis, scaled with the x-axis, is called abscissa or x coo ...
,
line
Line most often refers to:
* Line (geometry), object with zero thickness and curvature that stretches to infinity
* Telephone line, a single-user circuit on a telephone communication system
Line, lines, The Line, or LINE may also refer to:
Arts ...
and
half-plane of convergence of a Dirichlet series are analogous to
radius
In classical geometry, a radius ( : radii) of a circle or sphere is any of the line segments from its center to its perimeter, and in more modern usage, it is also their length. The name comes from the latin ''radius'', meaning ray but also the ...
,
boundary
Boundary or Boundaries may refer to:
* Border, in political geography
Entertainment
*Boundaries (2016 film), ''Boundaries'' (2016 film), a 2016 Canadian film
*Boundaries (2018 film), ''Boundaries'' (2018 film), a 2018 American-Canadian road trip ...
and
disk
Disk or disc may refer to:
* Disk (mathematics), a geometric shape
* Disk storage
Music
* Disc (band), an American experimental music band
* ''Disk'' (album), a 1995 EP by Moby
Other uses
* Disk (functional analysis), a subset of a vector sp ...
of convergence of a
power series
In mathematics, a power series (in one variable) is an infinite series of the form
\sum_^\infty a_n \left(x - c\right)^n = a_0 + a_1 (x - c) + a_2 (x - c)^2 + \dots
where ''an'' represents the coefficient of the ''n''th term and ''c'' is a const ...
.
On the line of convergence, the question of convergence remains open as in the case of power series. However, if a Dirichlet series converges and diverges at different points on the same vertical line, then this line must be the line of convergence. The proof is implicit in the definition of abscissa of convergence. An example would be the series
:
which converges at
(
alternating harmonic series
In mathematics, the harmonic series is the infinite series formed by summing all positive unit fractions:
\sum_^\infty\frac = 1 + \frac + \frac + \frac + \frac + \cdots.
The first n terms of the series sum to approximately \ln n + \gamma, where ...
) and diverges at
(
harmonic series). Thus,
is the line of convergence.
Suppose that a Dirichlet series does not converge at
, then it is clear that
and
diverges. On the other hand, if a Dirichlet series converges at
, then
and
converges. Thus, there are two formulas to compute
, depending on the convergence of
which can be determined by various
convergence tests
In mathematics, convergence tests are methods of testing for the convergence, conditional convergence, absolute convergence, interval of convergence or divergence of an infinite series \sum_^\infty a_n.
List of tests
Limit of the summand
If t ...
. These formulas are similar to the
Cauchy–Hadamard theorem
In mathematics, the Cauchy–Hadamard theorem is a result in complex analysis named after the French mathematicians Augustin Louis Cauchy and Jacques Hadamard, describing the radius of convergence of a power series. It was published in 1821 by C ...
for the radius of convergence of a power series.
If
is divergent, i.e.
, then
is given by
:
If
is convergent, i.e.
, then
is given by
:
Abscissa of absolute convergence
A Dirichlet series is
absolutely convergent
In mathematics, an infinite series of numbers is said to converge absolutely (or to be absolutely convergent) if the sum of the absolute values of the summands is finite. More precisely, a real or complex series \textstyle\sum_^\infty a_n is said ...
if the series
:
is convergent. As usual, an absolutely convergent Dirichlet series is convergent, but the
converse
Converse may refer to:
Mathematics and logic
* Converse (logic), the result of reversing the two parts of a definite or implicational statement
** Converse implication, the converse of a material implication
** Converse nonimplication, a logical c ...
is not always true.
If a Dirichlet series is absolutely convergent at
, then it is absolutely convergent for all ''s'' where
. A Dirichlet series may converge absolutely for all, for no or for some values of ''s''. In the latter case, there exist a
such that the series converges absolutely for
and converges non-absolutely for
.
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. There are also two formulas to compute
.
If
is divergent, then
is given by
:
If
is convergent, then
is given by
:
In general, the abscissa of convergence does not coincide with abscissa of absolute convergence. Thus, there might be a strip between the line of convergence and absolute convergence where a Dirichlet series is
conditionally convergent In mathematics, a series or integral is said to be conditionally convergent if it converges, but it does not converge absolutely.
Definition
More precisely, a series of real numbers \sum_^\infty a_n is said to converge conditionally if
\lim_\,\su ...
. 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
In mathematics, a Dirichlet series is any series of the form
\sum_^\infty \frac,
where ''s'' is complex, and a_n is a complex sequence. It is a special case of general Dirichlet series.
Dirichlet series play a variety of important roles in analyti ...
by substituting
.
Other abscissas of convergence
It is possible to consider other abscissas of convergence for a Dirichlet series. The abscissa of bounded convergence
is given by
while the abscissa of uniform convergence
is given by
These abscissas are related to the abscissa of convergence
and of absolute convergence
by the formulas
,
and a remarkable theorem of Bohr in fact shows that for any ordinary Dirichlet series where
(i.e. Dirichlet series of the form
) ,
and
Bohnenblust and Hille subsequently showed that for every number