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General Dirichlet series
In the field of mathematical analysis, a general Dirichlet series is an infinite series that takes the form of
\( \sum_{n=1}^{\infty}a_n e^{-\lambda_n s}, \)
where \(a_n, s are complex numbers and \( \{\lambda_n\} is a strictly increasing sequence of positive numbers that tends to infinity.
A simple observation shows that an 'ordinary' Dirichlet series
\( \sum_{n=1}^{\infty}\frac{a_n}{n^s}, \)
is obtained by substituting \lambda_n=\log n while a power series
\( \sum_{n=1}^{\infty}a_n (e^{-s})^n, \)
is obtained when \( \lambda_n=n \).
Fundamental theorems
If a Dirichlet series is convergent at s_0=\sigma_0+t_0i, then it is uniformly convergent in the domain
\( |\text{arg}(s-s_0)|\leq\theta<\frac{\pi}{2}, \)
and convergent for any \( s=\sigma+ti \) where \( \sigma>\sigma_0 \).
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 \(\sigma_c \) such that the series is convergent for \(\sigma>\sigma_c \) and divergent for\( \sigma<\sigma_c. By convention, \(\sigma_c=\infty \) if the series converges nowhere and \( \sigma_c=-\infty \) if the series converges everywhere on the complex plane.
Abscissa of convergence
The abscissa of convergence of a Dirichlet series can be defined as \( \sigma_c \)above. Another equivalent definition is
\( \sigma_c=\inf\{\sigma\in\mathbb{R}:\sum_{n=1}^{\infty}a_n e^{-\lambda_n s} \text{ converges for any } s \text{ where Re}(s)>\sigma\}. \)
The line \( \sigma=\sigma_c is called the line of convergence. The half-plane of convergence is defined as
\( \mathbb{C}_{\sigma_c}=\{s\in\mathbb{C}: \text{Re}(s)>\sigma_c\}. \)
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
\( \sum_{n=1}^{\infty}\frac{1}{n}e^{-ns}, \)
which converges at s=-\pi i (alternating harmonic series) and diverges at s=0 (harmonic series). Thus, \sigma=0 is the line of convergence.
Suppose that a Dirichlet series does not converge at s=0, then it is clear that \(\sigma_c\geq0 and \sum a_n diverges. On the other hand, if a Dirichlet series converges at s=0, then \sigma_c\leq0 and \sum a_n converges. Thus, there are two formulas to compute \( \sigma_c, depending on the convergence of\( \sum a_n 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 \( \sum a_k \) is divergent, i.e. \( \sigma_c\geq0 \), then \( \sigma_c \) is given by
\( \sigma_c=\limsup_{n\to\infty}\frac{\log|a_1+a_2+\cdots+a_n|}{\lambda_n}. \)
If \(\sum a_k \) is convergent, i.e. \(\sigma_c\leq0 \), then \( \sigma_c \) is given by
\( \sigma_c=\limsup_{n\to\infty}\frac{\log|a_{n+1}+a_{n+2}+\cdots|}{\lambda_n}. \)
Abscissa of absolute convergence
A Dirichlet series is absolutely convergent if the series
\( \sum_{n=1}^{\infty}|a_n e^{-\lambda_n s}|, \)
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 s_0, then it is absolutely convergent for all s where \( \text{Re}(s)>\text{Re}(s_0) \). A Dirichlet series may converge absolutely for all, for no or for some values of s. In the latter case, there exist a \sigma_a such that the series converges absolutely for \(\sigma>\sigma_a \) and converges non-absolutely for \( \sigma<\sigma_a \).
The abscissa of absolute convergence can be defined as \sigma_a above, or equivalently as
\( \sigma_a=\inf\{\sigma\in\mathbb{R}:\sum_{n=1}^{\infty}a_n e^{-\lambda_n s} \text{ converges absolutely for any } s \text{ where Re}(s)>\sigma\}. \)
The line and half-plane of absolute convergence can be defined similarly. There are also two formulas to compute \( \sigma_a \).
If \( \sum |a_k| \) is divergent, then \( \sigma_a \) is given by
\( \sigma_a=\limsup_{n\to\infty}\frac{\log(|a_1|+|a_2|+\cdots+|a_n|)}{\lambda_n}. \)
If \( \sum |a_k| \) is convergent, then \sigma_a is given by
\( \sigma_a=\limsup_{n\to\infty}\frac{\log(|a_{n+1}|+|a_{n+2}|+\cdots)}{\lambda_n}. \)
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
\( 0\leq\sigma_a-\sigma_c\leq L:=\limsup_{n\to\infty}\frac{\log n}{\lambda_n}. \)
In the case where L= 0, then
\( \sigma_c=\sigma_a=\limsup_{n\to\infty}\frac{\log |a_n|}{\lambda_n}. \)
All the formulas provided so far still hold true for 'ordinary' Dirichlet series by substituting \lambda_n=\log n.
Analytic functions
A function represented by a Dirichlet series
\( f(s)=\sum_{n=1}^{\infty}a_n e^{-\lambda_n s}, \)
is analytic on the half-plane of convergence. Moreover, for k=1,2,3,...
\( f^{(k)}(s)=(-1)^k\sum_{n=1}^{\infty}a_n\lambda_n^k e^{-\lambda_n s}. \)
Further generalizations
A Dirichlet series can be further generalized to the multi-variable case where \( \lambda_n\in\mathbb{R}^k, k = 2, 3, 4,..., \) or complex variable case where \( \lambda_n\in\mathbb{C}^m, m = 1, 2, 3,... \)
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
Dirichlet series at PlanetMath.org.
Hazewinkel, Michiel, ed. (2001), "Dirichlet series", Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
Undergraduate Texts in Mathematics
Graduate Studies in Mathematics
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