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$$ \tag{1 }
 
$$ \tag{1 }
\sum _ { n= } 1 ^  \infty  a _ {n} e ^ {- \lambda _ {n} s } ,
+
\sum _ { n=1 } ^  \infty  a _ {n} e ^ {- \lambda _ {n} s } ,
 
$$
 
$$
  
Line 25: Line 21:
  
 
$$  
 
$$  
\sum _ { n= } 1 ^  \infty   
+
\sum _ { n=1 } ^  \infty   
 
\frac{a _ {n} }{n  ^ {s} }
 
\frac{a _ {n} }{n  ^ {s} }
 
  .
 
  .
Line 33: Line 29:
  
 
$$  
 
$$  
\sum _ { n= } 1 ^  \infty   
+
\sum _ { n=1 } ^  \infty   
 
\frac{1}{n  ^ {s} }
 
\frac{1}{n  ^ {s} }
  
Line 42: Line 38:
  
 
$$  
 
$$  
L ( s)  =  \sum _ { n= } 1 ^  \infty   
+
L (s)  =  \sum _ { n=1 } ^  \infty   
\frac{\chi ( n) }{n  ^ {s} }
+
\frac{\chi (n) }{n  ^ {s} }
 
  ,
 
  ,
 
$$
 
$$
  
where  $  \chi ( n) $
+
where  $  \chi (n) $
 
is a function, known as a [[Dirichlet character|Dirichlet character]], were studied by P.G.L. Dirichlet (cf. [[Dirichlet L-function|Dirichlet  $  L $-
 
is a function, known as a [[Dirichlet character|Dirichlet character]], were studied by P.G.L. Dirichlet (cf. [[Dirichlet L-function|Dirichlet  $  L $-
 
function]]). Series (1) with arbitrary exponents  $  \lambda _ {n} $
 
function]]). Series (1) with arbitrary exponents  $  \lambda _ {n} $
Line 72: Line 68:
 
$$
 
$$
  
and there exist Dirichlet series for which  $  a- c = d $.  
+
and there exist Dirichlet series for which  $  a-c = d $.  
If  $  d= 0 $,  
+
If  $  d=0 $,  
 
the abscissa of convergence (abscissa of absolute convergence) is computed by the formula
 
the abscissa of convergence (abscissa of absolute convergence) is computed by the formula
  
Line 82: Line 78:
 
$$
 
$$
  
which is the analogue of the Cauchy–Hadamard formula. The case  $  d> 0 $
+
which is the analogue of the Cauchy–Hadamard formula. The case  $  d>0 $
 
is more complicated: If the magnitude
 
is more complicated: If the magnitude
  
Line 89: Line 85:
 
\frac{1}{\lambda _ {n} }
 
\frac{1}{\lambda _ {n} }
  
  \mathop{\rm ln}  \left | \sum _ { i= } 1 ^ { n }  a _ {i} \right |
+
  \mathop{\rm ln}  \left | \sum _ { i=1 } ^ { n }  a _ {i} \right |
 
$$
 
$$
  
Line 95: Line 91:
 
if  $  \beta \leq  0 $
 
if  $  \beta \leq  0 $
 
and the series (1) diverges at the point  $  s = 0 $,  
 
and the series (1) diverges at the point  $  s = 0 $,  
then  $  c= 0 $;  
+
then  $  c=0 $;  
 
if  $  \beta \leq  0 $
 
if  $  \beta \leq  0 $
 
and the series (1) converges at the point  $  s = 0 $,  
 
and the series (1) converges at the point  $  s = 0 $,  
Line 104: Line 100:
 
\frac{1}{\lambda _ {n} }
 
\frac{1}{\lambda _ {n} }
  
  \mathop{\rm ln}  \left | \sum _ { i= } 1 ^  \infty  a _ {i} \right | .
+
  \mathop{\rm ln}  \left | \sum _ { i=1 } ^  \infty  a _ {i} \right | .
 
$$
 
$$
  
The sum of the series,  $  F ( s) $,  
+
The sum of the series,  $  F (s) $,  
 
is an analytic function in the half-plane of convergence. If  $  \sigma \rightarrow + \infty $,  
 
is an analytic function in the half-plane of convergence. If  $  \sigma \rightarrow + \infty $,  
 
the function  $  F ( \sigma ) $
 
the function  $  F ( \sigma ) $
Line 113: Line 109:
 
if  $  a _ {1} \neq 0 $).  
 
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 $
 
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) $
+
in which  $  F (s) $
is an analytic function is said to be the half-plane of holomorphy of the function  $  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 $
 
the straight line  $  \sigma = h $
 
is known as the axis of holomorphy and the number  $  h $
 
is known as the axis of holomorphy and the number  $  h $
is called the abscissa of holomorphy. The inequality  $  h\leq c $
+
is called the abscissa of holomorphy. The inequality  $  h\leq c $
is true, and cases when  $  h< c $
+
is true, and cases when  $  h<c $
 
are possible. Let  $  q $
 
are possible. Let  $  q $
 
be the greatest lower bound of the numbers  $  \beta $
 
be the greatest lower bound of the numbers  $  \beta $
for which  $  F ( s) $
+
for which  $  F (s) $
 
is bounded in modulus in the half-plane  $  \sigma > \beta $(
 
is bounded in modulus in the half-plane  $  \sigma > \beta $(
 
$  q \leq  a $).  
 
$  q \leq  a $).  
Line 129: Line 125:
 
a _ {n}  =  \lim\limits _ {T \rightarrow \infty }   
 
a _ {n}  =  \lim\limits _ {T \rightarrow \infty }   
 
\frac{1}{2T}
 
\frac{1}{2T}
  \int\limits _ { p- } iT ^ { p+ }  iT F ( s) e ^ {\lambda _ {n} s }  ds,\  n= 1, 2 \dots p> q,
+
  \int\limits _ { p-iT } ^ { p+iT }  F (s) e ^ {\lambda _ {n} s }  ds,\  n=1, 2 \dots p>q,
 
$$
 
$$
  
Line 146: Line 142:
 
The sum of a Dirichlet series cannot be an arbitrary analytic function in some half-plane  $  \sigma > h $;  
 
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 $.  
 
it must, for example, tend to zero if  $  \sigma \rightarrow + \infty $.  
However, the following holds: Whatever the analytic function  $  \phi ( s) $
+
However, the following holds: Whatever the analytic function  $  \phi (s) $
 
in the half-plane  $  \sigma > h $,  
 
in the half-plane  $  \sigma > h $,  
it is possible to find a Dirichlet series (1) such that its sum  $  F ( s) $
+
it is possible to find a Dirichlet series (1) such that its sum  $  F (s) $
will differ from  $  \phi ( s) $
+
will differ from  $  \phi (s) $
 
by an entire function.
 
by an entire function.
  
Line 179: Line 175:
 
The value of  $  \delta $
 
The value of  $  \delta $
 
may be arbitrary in  $  [ 0 , \infty ] $;  
 
may be arbitrary in  $  [ 0 , \infty ] $;  
in particular, if  $  \lambda _ {n+} 1 - \lambda _ {n} \geq  q > 0 $,  
+
in particular, if  $  \lambda _ {n+1} - \lambda _ {n} \geq  q > 0 $,  
 
$  n = 1 , 2 \dots $
 
$  n = 1 , 2 \dots $
 
then  $  \delta = 0 $.  
 
then  $  \delta = 0 $.  
Line 185: Line 181:
 
the sum of the series has at least one singular point.
 
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) $
+
If the Dirichlet series (1) converges in the entire plane, its sum  $  F (s) $
 
is an entire function. Let
 
is an entire function. Let
  
Line 195: Line 191:
 
$$
 
$$
  
then the R-order of the entire function  $  F ( s) $(
+
then the R-order of the entire function  $  F (s) $(
 
Ritt order) is the magnitude
 
Ritt order) is the magnitude
  
Line 216: Line 212:
 
$$
 
$$
  
One can also introduce the concept of the R-type of a function  $  F ( s) $.
+
One can also introduce the concept of the R-type of a function  $  F (s) $.
  
 
If
 
If
Line 227: Line 223:
 
$$
 
$$
  
and if the function  $  F ( s) $
+
and if the function  $  F (s) $
 
is bounded in modulus in a horizontal strip wider than  $  2 \pi \tau $,  
 
is bounded in modulus in a horizontal strip wider than  $  2 \pi \tau $,  
then  $  F ( s) \equiv 0 $(
+
then  $  F (s) \equiv 0 $(
 
the analogue of one of the [[Liouville theorems|Liouville theorems]]).
 
the analogue of one of the [[Liouville theorems|Liouville theorems]]).
  
Line 236: Line 232:
  
 
$$ \tag{2 }
 
$$ \tag{2 }
F ( s)  =  \sum _ {n = 1 } ^  \infty  a _ {n} e ^ {- \lambda _ {n} s }
+
F (s)  =  \sum _ {n = 1 } ^  \infty  a _ {n} e ^ {- \lambda _ {n} s }
 
$$
 
$$
  
Line 249: Line 245:
 
$$
 
$$
  
the open domains of convergence and absolute convergence coincide. The sum  $  F ( s) $
+
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) $
+
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
 
is, generally speaking, wider than the domain of convergence of the Dirichlet series (2). If
  
Line 273: Line 269:
 
be an entire function of exponential type which has simple zeros at the points  $  \lambda _ {n} $,  
 
be an entire function of exponential type which has simple zeros at the points  $  \lambda _ {n} $,  
 
$  n \geq  1 $;  
 
$  n \geq  1 $;  
let  $  \gamma ( t) $
+
let  $  \gamma (t) $
 
be the Borel-associated function to  $  L ( \lambda ) $(
 
be the Borel-associated function to  $  L ( \lambda ) $(
 
cf. [[Borel transform|Borel transform]]); let  $  \overline{D}\; $
 
cf. [[Borel transform|Borel transform]]); let  $  \overline{D}\; $
be the smallest closed convex set containing all the singular points of  $  \gamma ( t) $,  
+
be the smallest closed convex set containing all the singular points of  $  \gamma (t) $,  
 
and let
 
and let
  
 
$$  
 
$$  
\psi _ {n} ( t)  =   
+
\psi _ {n} (t)  =   
 
\frac{1}{L  ^  \prime  ( \lambda _ {n} ) }
 
\frac{1}{L  ^  \prime  ( \lambda _ {n} ) }
  
Line 289: Line 285:
 
$$
 
$$
  
Then the functions  $  \psi _ {n} ( t) $
+
Then the functions  $  \psi _ {n} (t) $
 
are regular outside  $  \overline{D}\; $,  
 
are regular outside  $  \overline{D}\; $,  
 
$  \psi _ {n} ( \infty ) = 0 $,  
 
$  \psi _ {n} ( \infty ) = 0 $,  
Line 297: Line 293:
  
 
\frac{1}{2 \pi i }
 
\frac{1}{2 \pi i }
  \int\limits _ { C } e ^ {\lambda _ {m} t } \psi _ {n} ( t)
+
  \int\limits _ { C } e ^ {\lambda _ {m} t } \psi _ {n} (t)
 
  d t  =  \left \{  
 
  d t  =  \left \{  
 
\begin{array}{ll}
 
\begin{array}{ll}
 
0 ,  & m \neq n ,  \\
 
0 ,  & m \neq n ,  \\
1,  & m = n ,  \\
+
1,  & m =n ,  \\
 
\end{array}
 
\end{array}
  
Line 308: Line 304:
 
where  $  C $
 
where  $  C $
 
is a closed contour encircling  $  \overline{D}\; $.  
 
is a closed contour encircling  $  \overline{D}\; $.  
If the functions  $  \psi _ {n} ( t) $
+
If the functions  $  \psi _ {n} (t) $
 
are continuous up to the boundary of  $  \overline{D}\; $,  
 
are continuous up to the boundary of  $  \overline{D}\; $,  
 
the boundary  $  \partial  \overline{D}\; $
 
the boundary  $  \partial  \overline{D}\; $
 
may be taken as  $  C $.  
 
may be taken as  $  C $.  
To an arbitrary analytic function  $  F ( s) $
+
To an arbitrary analytic function  $  F (s) $
 
in  $  D $(
 
in  $  D $(
 
the interior of the domain  $  \overline{D}\; $)  
 
the interior of the domain  $  \overline{D}\; $)  
Line 319: Line 315:
  
 
$$ \tag{3 }
 
$$ \tag{3 }
F ( s)  \sim  \sum _ {n = 1 } ^  \infty   
+
F (s)  \sim  \sum _ {n = 1 } ^  \infty   
 
a _ {n} e ^ {\lambda _ {n} s } ,
 
a _ {n} e ^ {\lambda _ {n} s } ,
 
$$
 
$$
Line 326: Line 322:
 
a _ {n}  =   
 
a _ {n}  =   
 
\frac{1}{2 \pi i }
 
\frac{1}{2 \pi i }
  \int\limits _ {\partial  D
+
  \int\limits _ {\partial  \overline{D}\; } F (t) \psi _ {n} (t)  d t ,\  n \geq  1 .
bar } F ( t) \psi _ {n} ( t)  d t ,\  n \geq  1 .
 
 
$$
 
$$
  
Line 333: Line 328:
 
it is possible to construct an entire function  $  L ( \lambda ) $
 
it is possible to construct an entire function  $  L ( \lambda ) $
 
with simple zeros  $  \lambda _ {1} , \lambda _ {2} \dots $
 
with simple zeros  $  \lambda _ {1} , \lambda _ {2} \dots $
such that for any function  $  F ( s) $
+
such that for any function  $  F (s) $
 
analytic in  $  D $
 
analytic in  $  D $
 
and continuous in  $  \overline{D}\; $
 
and continuous in  $  \overline{D}\; $
 
the series (3) converges uniformly inside  $  D $
 
the series (3) converges uniformly inside  $  D $
to  $  F ( s) $.  
+
to  $  F (s) $.  
For an analytic function  $  \phi ( s) $
+
For an analytic function  $  \phi (s) $
 
in  $  D $(
 
in  $  D $(
 
not necessarily continuous in  $  \overline{D}\; $)  
 
not necessarily continuous in  $  \overline{D}\; $)  
Line 347: Line 342:
 
$$
 
$$
  
and a function  $  F ( s) $
+
and a function  $  F (s) $
 
analytic in  $  D $
 
analytic in  $  D $
 
and continuous in  $  \overline{D}\; $,  
 
and continuous in  $  \overline{D}\; $,  
Line 353: Line 348:
  
 
$$  
 
$$  
\phi ( s)  =  M ( D ) F ( s)  =  \sum _ {n= 0 } ^  \infty  c _ {n} F ^ { ( n) } ( s) .
+
\phi (s)  =  M ( D ) F (s)  =  \sum _ {n=0 } ^  \infty  c _ {n} F ^ { (n) } (s) .
 
$$
 
$$
  
Line 359: Line 354:
  
 
$$  
 
$$  
\phi ( s)  =  \sum _ {n = 0 } ^  \infty  a _ {n} M ( \lambda _ {n} )
+
\phi (s)  =  \sum _ {n = 0 } ^  \infty  a _ {n} M ( \lambda _ {n} )
 
e ^ {\lambda _ {n} s } ,\  s \in D .
 
e ^ {\lambda _ {n} s } ,\  s \in D .
 
$$
 
$$

Revision as of 22:06, 30 December 2020

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

(Automatically converted into $\TeX$. Above some diagnostics.)

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).

References

[1] A.F. Leont'ev, "Exponential series" , Moscow (1976) (In Russian)
[2] S. Mandelbrojt, "Dirichlet series, principles and methods" , Reidel (1972)

Comments

References

[a1] G.H. Hardy, M. Riesz, "The general theory of Dirichlet series" , Cambridge Univ. Press (1915) Zbl 45.0387.03
How to Cite This Entry:
Dirichlet series. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Dirichlet_series&oldid=46723
This article was adapted from an original article by A.F. Leont'ev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article