# Dirichlet series

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d0329201.png $#A+1 = 131 n = 0$#C+1 = 131 : ~/encyclopedia/old_files/data/D032/D.0302920 Dirichlet series

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A series of the form

$$\tag{1 } \sum _ { n=1 } ^ \infty a _ {n} e ^ {- \lambda _ {n} s } ,$$

where the $a _ {n}$ are complex coefficients, $\lambda _ {n}$, $0 < | \lambda _ {n} | \uparrow \infty$, are the exponents of the series, and $s = \sigma + it$ is a complex variable. If $\lambda _ {n} = \mathop{\rm ln} n$, one obtains the so-called ordinary Dirichlet series

$$\sum _ { n=1 } ^ \infty \frac{a _ {n} }{n ^ {s} } .$$

The series

$$\sum _ { n=1 } ^ \infty \frac{1}{n ^ {s} }$$

represents the Riemann zeta-function for $\sigma > 1$. The series

$$L (s) = \sum _ { n=1 } ^ \infty \frac{\chi (n) }{n ^ {s} } ,$$

where $\chi (n)$ is a function, known as a Dirichlet character, were studied by P.G.L. Dirichlet (cf. Dirichlet $L$- function). Series (1) with arbitrary exponents $\lambda _ {n}$ are known as general Dirichlet series.

## General Dirichlet series with positive exponents.

Let, initially, the $\lambda _ {n}$ be positive numbers. The analogue of the Abel theorem for power series is then valid: If the series (1) converges at a point $s _ {0} = \sigma _ {0} + it _ {0}$, it will converge in the half-plane $\sigma > \sigma _ {0}$, and it will converge uniformly inside an arbitrary angle $| \mathop{\rm arg} ( s - s _ {0} ) | < \phi _ {0} < \pi / 2$. The open domain of convergence of the series is some half-plane $\sigma > c$. The number $c$ is said to be the abscissa of convergence of the Dirichlet series; the straight line $\sigma = c$ is said to be the axis of convergence of the series, and the half-plane $\sigma > c$ is said to be the half-plane of convergence of the series. As well as the half-plane of convergence one also considers the half-plane of absolute convergence of the Dirichlet series, $\sigma > a$: The open domain in which the series converges absolutely (here $a$ is the abscissa of absolute convergence). In general, the abscissas of convergence and of absolute convergence are different. But always:

$$0 \leq a - c \leq d ,\ \textrm{ where } d = \overline{\lim\limits}\; _ {n\rightarrow \infty } \ \frac{ \mathop{\rm ln} n }{\lambda _ {n} } ,$$

and there exist Dirichlet series for which $a-c = d$. If $d=0$, the abscissa of convergence (abscissa of absolute convergence) is computed by the formula

$$a = c = \overline{\lim\limits}\; _ {n \rightarrow \infty } \frac{ \mathop{\rm ln} | a _ {n} | }{\lambda _ {n} } ,$$

which is the analogue of the Cauchy–Hadamard formula. The case $d>0$ is more complicated: If the magnitude

$$\beta = \overline{\lim\limits}\; _ {n \rightarrow \infty } \frac{1}{\lambda _ {n} } \mathop{\rm ln} \left | \sum _ { i=1 } ^ { n } a _ {i} \right |$$

is positive, then $c = \beta$; if $\beta \leq 0$ and the series (1) diverges at the point $s = 0$, then $c=0$; if $\beta \leq 0$ and the series (1) converges at the point $s = 0$, then

$$c = \overline{\lim\limits}\; _ {n \rightarrow \infty } \frac{1}{\lambda _ {n} } \mathop{\rm ln} \left | \sum _ { i=1 } ^ \infty a _ {i} \right | .$$

The sum of the series, $F (s)$, is an analytic function in the half-plane of convergence. If $\sigma \rightarrow + \infty$, the function $F ( \sigma )$ asymptotically behaves as the first term of the series, $a _ {1} e ^ {- \lambda _ {1} \sigma }$( if $a _ {1} \neq 0$). If the sum of the series is zero, then all coefficients of the series are zero. The maximal half-plane $\sigma > h$ in which $F (s)$ is an analytic function is said to be the half-plane of holomorphy of the function $F (s)$, the straight line $\sigma = h$ is known as the axis of holomorphy and the number $h$ is called the abscissa of holomorphy. The inequality $h\leq c$ is true, and cases when $h<c$ are possible. Let $q$ be the greatest lower bound of the numbers $\beta$ for which $F (s)$ is bounded in modulus in the half-plane $\sigma > \beta$( $q \leq a$). The formula

$$a _ {n} = \lim\limits _ {T \rightarrow \infty } \frac{1}{2T} \int\limits _ { p-iT } ^ { p+iT } F (s) e ^ {\lambda _ {n} s } ds,\ n=1, 2 \dots p>q,$$

is valid, and entails the inequalities

$$| a _ {n} | \leq \frac{M ( \sigma ) }{e ^ {- \lambda _ {n} \sigma } } ,\ M ( \sigma ) = \sup _ {- \infty < t < \infty } | F ( \sigma + it ) | ,$$

which are analogues of the Cauchy inequalities for the coefficients of a power series.

The sum of a Dirichlet series cannot be an arbitrary analytic function in some half-plane $\sigma > h$; it must, for example, tend to zero if $\sigma \rightarrow + \infty$. However, the following holds: Whatever the analytic function $\phi (s)$ in the half-plane $\sigma > h$, it is possible to find a Dirichlet series (1) such that its sum $F (s)$ will differ from $\phi (s)$ by an entire function.

If the sequence of exponents has a density

$$\tau = \lim\limits _ {n \rightarrow \infty } \ \frac{n}{\lambda _ {n} } < \infty ,$$

the difference between the abscissa of convergence (the abscissas of convergence and of absolute convergence coincide) and the abscissa of holomorphy does not exceed

$$\delta = \overline{\lim\limits}\; _ {n \rightarrow \infty } \frac{1}{\lambda _ {n} } \mathop{\rm ln} \left | \frac{1}{L ^ \prime ( \lambda _ {n} ) } \right | ,\ \ L ( \lambda ) = \prod _ {n = 1 } ^ \infty \left ( 1 - \frac{\lambda ^ {2} }{\lambda _ {n} ^ {2} } \right ) ,$$

and there exist series for which this difference equals $\delta$. The value of $\delta$ may be arbitrary in $[ 0 , \infty ]$; in particular, if $\lambda _ {n+1} - \lambda _ {n} \geq q > 0$, $n = 1 , 2 \dots$ then $\delta = 0$. The axis of holomorphy has the following property: On any of its segments of length $2 \pi \tau$ the sum of the series has at least one singular point.

If the Dirichlet series (1) converges in the entire plane, its sum $F (s)$ is an entire function. Let

$$\overline{\lim\limits}\; _ {n \rightarrow \infty } \ \frac{ \mathop{\rm ln} n }{\lambda _ {n} } < \infty ;$$

then the R-order of the entire function $F (s)$( Ritt order) is the magnitude

$$\rho = \overline{\lim\limits}\; _ {\sigma \rightarrow - \infty } \ \frac{ { \mathop{\rm ln} \mathop{\rm ln} } M ( \sigma ) }{- \sigma } .$$

Its expression in terms of the coefficients of the series is

$$- \frac{1} \rho = \overline{\lim\limits}\; _ {n \rightarrow \infty } \ \frac{ \mathop{\rm ln} | a _ {n} | }{\lambda _ {n} \mathop{\rm ln} \lambda _ {n} } .$$

One can also introduce the concept of the R-type of a function $F (s)$.

If

$$\overline{\lim\limits}\; _ {n \rightarrow \infty } \frac{n}{\lambda _ {n} } = \ \tau < \infty$$

and if the function $F (s)$ is bounded in modulus in a horizontal strip wider than $2 \pi \tau$, then $F (s) \equiv 0$( the analogue of one of the Liouville theorems).

## Dirichlet series with complex exponents.

For a Dirichlet series

$$\tag{2 } F (s) = \sum _ {n = 1 } ^ \infty a _ {n} e ^ {- \lambda _ {n} s }$$

with complex exponents $0 < | \lambda _ {1} | \leq | \lambda _ {2} | \leq \dots$, the open domain of absolute convergence is convex. If

$$\lim\limits _ {n \rightarrow \infty } \ \frac{ \mathop{\rm ln} n }{\lambda _ {n} } = 0 ,$$

the open domains of convergence and absolute convergence coincide. The sum $F (s)$ of the series (2) is an analytic function in the domain of convergence. The domain of holomorphy of $F (s)$ is, generally speaking, wider than the domain of convergence of the Dirichlet series (2). If

$$\lim\limits _ {n \rightarrow \infty } \frac{n}{\lambda _ {n} } = 0,$$

then the domain of holomorphy is convex.

Let

$$\overline{\lim\limits}\; _ {n \rightarrow \infty } \frac{n}{| \lambda _ {n} | } = \tau < \infty ;$$

let $L ( \lambda )$ be an entire function of exponential type which has simple zeros at the points $\lambda _ {n}$, $n \geq 1$; let $\gamma (t)$ be the Borel-associated function to $L ( \lambda )$( cf. Borel transform); let $\overline{D}\;$ be the smallest closed convex set containing all the singular points of $\gamma (t)$, and let

$$\psi _ {n} (t) = \frac{1}{L ^ \prime ( \lambda _ {n} ) } \int\limits _ { 0 } ^ \infty \frac{L ( \lambda ) }{\lambda - \lambda _ {n} } e ^ {- \lambda t } d \lambda ,\ n = 1 , 2 , . . . .$$

Then the functions $\psi _ {n} (t)$ are regular outside $\overline{D}\;$, $\psi _ {n} ( \infty ) = 0$, and they are bi-orthogonal to the system $\{ e ^ {\lambda _ {n} s } \}$:

$$\frac{1}{2 \pi i } \int\limits _ { C } e ^ {\lambda _ {m} t } \psi _ {n} (t) d t = \left \{ \begin{array}{ll} 0 , & m \neq n , \\ 1, & m =n , \\ \end{array} \right .$$

where $C$ is a closed contour encircling $\overline{D}\;$. If the functions $\psi _ {n} (t)$ are continuous up to the boundary of $\overline{D}\;$, the boundary $\partial \overline{D}\;$ may be taken as $C$. To an arbitrary analytic function $F (s)$ in $D$( the interior of the domain $\overline{D}\;$) which is continuous in $\overline{D}\;$ one assigns a series:

$$\tag{3 } F (s) \sim \sum _ {n = 1 } ^ \infty a _ {n} e ^ {\lambda _ {n} s } ,$$

$$a _ {n} = \frac{1}{2 \pi i } \int\limits _ {\partial \overline{D}\; } F (t) \psi _ {n} (t) d t ,\ n \geq 1 .$$

For a given bounded convex domain $\overline{D}\;$ it is possible to construct an entire function $L ( \lambda )$ with simple zeros $\lambda _ {1} , \lambda _ {2} \dots$ such that for any function $F (s)$ analytic in $D$ and continuous in $\overline{D}\;$ the series (3) converges uniformly inside $D$ to $F (s)$. For an analytic function $\phi (s)$ in $D$( not necessarily continuous in $\overline{D}\;$) it is possible to find an entire function of exponential type zero,

$$M ( \lambda ) = \sum _ {n = 0 } ^ \infty c _ {n} \lambda ^ {n} ,$$

and a function $F (s)$ analytic in $D$ and continuous in $\overline{D}\;$, such that

$$\phi (s) = M ( D ) F (s) = \sum _ {n=0 } ^ \infty c _ {n} F ^ { (n) } (s) .$$

Then

$$\phi (s) = \sum _ {n = 0 } ^ \infty a _ {n} M ( \lambda _ {n} ) e ^ {\lambda _ {n} s } ,\ s \in D .$$

The representation of arbitrary analytic functions by Dirichlet series in a domain $D$ was also established in cases when $D$ is the entire plane or an infinite convex polygonal domain (bounded by a finite number of rectilinear segments).

How to Cite This Entry:
Abscissa of convergence. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Abscissa_of_convergence&oldid=37124