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In the field of [[mathematical analysis]], a '''general Dirichlet series''' is an [[series (mathematics)|infinite series]] that takes the form of | |||
: <math>\sum_{n=1}^{\infty}a_n e^{-\lambda_n s},</math> | |||
where <math>a_n</math>, <math>s</math> are [[complex number]]s and <math>\{\lambda_n\}</math> is a strictly increasing [[sequence]] of positive numbers that tends to infinity. | |||
A simple observation shows that an 'ordinary' [[Dirichlet series]] | |||
: <math>\sum_{n=1}^{\infty}\frac{a_n}{n^s},</math> | |||
is obtained by substituting <math>\lambda_n=\log n</math> while a [[power series]] | |||
: <math>\sum_{n=1}^{\infty}a_n (e^{-s})^n,</math> | |||
is obtained when <math>\lambda_n=n</math>. | |||
== Fundamental theorems == | |||
If a Dirichlet series is convergent at <math>s_0=\sigma_0+t_0i</math>, then it is [[uniform convergence|uniformly convergent]] in the [[Domain of a function|domain]] | |||
: <math>|\text{arg}(s-s_0)|\leq\theta<\frac{\pi}{2},</math> | |||
and [[Convergent series|convergent]] for any <math>s=\sigma+ti</math> where <math>\sigma>\sigma_0</math>. | |||
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 <math>\sigma_c</math> such that the series is convergent for <math>\sigma>\sigma_c</math> and [[divergent series|divergent]] for <math>\sigma<\sigma_c</math>. By convention, <math>\sigma_c=\infty</math> if the series converges nowhere and <math>\sigma_c=-\infty</math> if the series converges everywhere on the [[complex plane]]. | |||
== Abscissa of convergence == | |||
The '''abscissa of convergence''' of a Dirichlet series can be defined as <math>\sigma_c</math> above. Another equivalent definition is | |||
: <math>\sigma_c=\inf\{\sigma\in\mathbb{R}:\sum_{n=1}^{\infty}a_n e^{-\lambda_n s} \text{ converges for any } s \text{ where Re}(s)>\sigma\}</math>. | |||
The line <math>\sigma=\sigma_c</math> is called the '''line of convergence'''. The '''half-plane of convergence''' is defined as | |||
: <math>\mathbb{C}_{\sigma_c}=\{s\in\mathbb{C}: \text{Re}(s)>\sigma_c\}.</math> | |||
The [[abscissa]], [[line (geometry)|line]] and [[Half-space (geometry)|half-plane]] of convergence of a Dirichlet series are analogous to [[radius]], [[Boundary (topology)|boundary]] and [[disk (mathematics)|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. 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 | |||
: <math>\sum_{n=1}^{\infty}\frac{1}{n}e^{-ns},</math> | |||
which converges at <math>s=-\pi i</math> ([[harmonic series (mathematics)|alternating harmonic series]]) and diverges at <math>s=0</math> ([[harmonic series (mathematics)|harmonic series]]). Thus, <math>\sigma=0</math> is the line of convergence. | |||
Suppose that a Dirichlet series does not converge at <math>s=0</math>, then it is clear that <math>\sigma_c\geq0</math> and <math>\sum a_n</math> diverges. On the other hand, if a Dirichlet series converges at <math>s=0</math>, then <math>\sigma_c\leq0</math> and <math>\sum a_n</math> converges. Thus, there are two formulas to compute <math>\sigma_c</math>, depending on the convergence of <math>\sum a_n</math> which can be determined by various [[convergence tests]]. These formulas are similar to the [[Cauchy–Hadamard theorem|Cauchy-Hadamard theorem]] for the radius of convergence of a power series. | |||
If <math>\sum a_k</math> is divergent, i.e. <math>\sigma_c\geq0</math>, then <math>\sigma_c</math> is given by | |||
: <math>\sigma_c=\limsup_{n\to\infty}\frac{\log|a_1+a_2+\cdots+a_n|}{\lambda_n}.</math> | |||
If <math>\sum a_k</math> is convergent, i.e. <math>\sigma_c\leq0</math>, then <math>\sigma_c</math> is given by | |||
: <math>\sigma_c=\limsup_{n\to\infty}\frac{\log|a_{n+1}+a_{n+2}+\cdots|}{\lambda_n}.</math> | |||
== Abscissa of absolute convergence == | |||
A Dirichlet series is [[absolute convergence|absolutely convergent]] if the series | |||
: <math>\sum_{n=1}^{\infty}|a_n e^{-\lambda_n s}|,</math> | |||
is convergent. As usual, an absolutely convergent Dirichlet series is convergent, but the [[Theorem#Converse|converse]] is not always true. | |||
If a Dirichlet series is absolutely convergent at <math>s_0</math>, then it is absolutely convergent for all ''s'' where <math>\text{Re}(s)>\text{Re}(s_0)</math>. A Dirichlet series may converge absolutely for all, for no or for some values of ''s''. In the latter case, there exist a <math>\sigma_a</math> such that the series converges absolutely for <math>\sigma>\sigma_a</math> and converges non-absolutely for <math>\sigma<\sigma_a</math>. | |||
The '''abscissa of absolute convergence''' can be defined as <math>\sigma_a</math> above, or equivalently as | |||
: <math>\sigma_c=\inf\{\sigma\in\mathbb{R}:\sum_{n=1}^{\infty}a_n e^{-\lambda_n s} \text{ converges absolutely for any } s \text{ where Re}(s)>\sigma\}</math>. | |||
The '''line''' and '''half-plane of absolute convergence''' can be defined similarly. There are also two formulas to compute <math>\sigma_a</math>. | |||
If <math>\sum |a_k|</math> is divergent, then <math>\sigma_a</math> is given by | |||
: <math>\sigma_a=\limsup_{n\to\infty}\frac{\log(|a_1|+|a_2|+\cdots+|a_n|)}{\lambda_n}.</math> | |||
If <math>\sum |a_k|</math> is convergent, then <math>\sigma_a</math> is given by | |||
: <math>\sigma_a=\limsup_{n\to\infty}\frac{\log(|a_{n+1}|+|a_{n+2}|+\cdots)}{\lambda_n}.</math> | |||
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 [[conditional convergence|conditionally convergent]]. The width of this strip is given by | |||
: <math>0\leq\sigma_a-\sigma_c\leq L:=\limsup_{n\to\infty}\frac{\log n}{\lambda_n}.</math> | |||
In the case where ''L''= 0, then | |||
: <math>\sigma_c=\sigma_a=\limsup_{n\to\infty}\frac{\log |a_n|}{\lambda_n}.</math> | |||
All the formulas provided so far still hold true for 'ordinary' [[Dirichlet series]] by substituting <math>\lambda_n=\log n</math>. | |||
== Analytic functions == | |||
A [[Function (mathematics)|function]] represented by a Dirichlet series | |||
: <math>f(s)=\sum_{n=1}^{\infty}a_n e^{-\lambda_n s},</math> | |||
is [[Analytic function|analytic]] on the half-plane of convergence. Moreover, for <math>k=1,2,3,...</math> | |||
: <math>f^{(k)}(s)=(-1)^k\sum_{n=1}^{\infty}a_n\lambda_n^k e^{-\lambda_n s}.</math> | |||
== Further generalizations == | |||
A Dirichlet series can be further generalized to the [[variable (mathematics)|multi-variable]] case where <math>\lambda_n\in\mathbb{R}^k</math>, ''k'' = 2, 3, 4,..., or [[complex analysis|complex variable]] case where <math>\lambda_n\in\mathbb{C}^m</math>, ''m'' = 1, 2, 3,... | |||
== References == | |||
* [[G. H. Hardy]], and M. Riesz, ''The general theory of Dirichlet's series'', Cambridge University Press, first edition, 1915. | |||
* [[Edward Charles Titchmarsh|E. C. Titchmarsh]], ''The theory of functions'', Oxford University Press, second edition, 1939. | |||
* [[Tom M. Apostol|Tom Apostol]], ''Modular functions and Dirichlet series in number theory'', Springer, second edition, 1990. | |||
* A.F. Leont'ev, ''Entire functions and series of exponentials'' (in Russian), Nauka, first edition, 1982. | |||
* A.I. Markushevich, ''Theory of functions of a complex variables'' (translated from Russian), Chelsea Publishing Company, second edition, 1977. | |||
* [[J.-P. Serre]], ''A Course in Arithmetic'', Springer-Verlag, fifth edition, 1973. | |||
== External links == | |||
* {{planetmath reference|title=Dirichlet series|id=4764}} | |||
* {{Springer|title = Dirichlet series|id=d/d032920}} | |||
[[Category:Complex analysis]] | |||
[[Category:Mathematical series]] |
Revision as of 04:54, 5 June 2013
In the field of mathematical analysis, a general Dirichlet series is an infinite series that takes the form of
where , are complex numbers and is a strictly increasing sequence of positive numbers that tends to infinity.
A simple observation shows that an 'ordinary' Dirichlet series
is obtained by substituting while a power series
Fundamental theorems
If a Dirichlet series is convergent at , then it is uniformly convergent in the domain
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.
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, 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. 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) 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. These formulas are similar to the Cauchy-Hadamard theorem 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 if the series
is convergent. As usual, an absolutely convergent Dirichlet series is convergent, but the converse 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. 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 .
Analytic functions
A function represented by a Dirichlet series
is analytic on the half-plane of convergence. Moreover, for
Further generalizations
A Dirichlet series can be further generalized to the multi-variable case where , k = 2, 3, 4,..., or complex variable case where , m = 1, 2, 3,...
References
- G. H. Hardy, and M. Riesz, The general theory of Dirichlet's series, Cambridge University Press, first edition, 1915.
- E. C. Titchmarsh, The theory of functions, Oxford University Press, second edition, 1939.
- Tom Apostol, Modular functions and Dirichlet series in number theory, Springer, second edition, 1990.
- A.F. Leont'ev, Entire functions and series of exponentials (in Russian), Nauka, first edition, 1982.
- A.I. Markushevich, Theory of functions of a complex variables (translated from Russian), Chelsea Publishing Company, second edition, 1977.
- J.-P. Serre, A Course in Arithmetic, Springer-Verlag, fifth edition, 1973.
External links
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