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A series of the form
 
A series of the form
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032920/d0329201.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
+
$$ \tag{1 }
 +
\sum _ { n= } 1 ^  \infty  a _ {n} e ^ {- \lambda _ {n} s } ,
 +
$$
  
where the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032920/d0329202.png" /> are complex coefficients, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032920/d0329203.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032920/d0329204.png" />, are the exponents of the series, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032920/d0329205.png" /> is a complex variable. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032920/d0329206.png" />, one obtains the so-called ordinary Dirichlet series
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032920/d0329207.png" /></td> </tr></table>
+
$$
 +
\sum _ { n= } 1 ^  \infty 
 +
\frac{a _ {n} }{n  ^ {s} }
 +
.
 +
$$
  
 
The series
 
The series
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032920/d0329208.png" /></td> </tr></table>
+
$$
 +
\sum _ { n= } 1 ^  \infty 
 +
\frac{1}{n  ^ {s} }
 +
 
 +
$$
  
represents the Riemann [[Zeta-function|zeta-function]] for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032920/d0329209.png" />. The series
+
represents the Riemann [[Zeta-function|zeta-function]] for $  \sigma > 1 $.  
 +
The series
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032920/d03292010.png" /></td> </tr></table>
+
$$
 +
L ( s)  = \sum _ { n= } 1 ^  \infty 
 +
\frac{\chi ( n) }{n  ^ {s} }
 +
,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032920/d03292011.png" /> is a function, known as a [[Dirichlet character|Dirichlet character]], were studied by P.G.L. Dirichlet (cf. [[Dirichlet L-function|Dirichlet <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032920/d03292012.png" />-function]]). Series (1) with arbitrary exponents <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032920/d03292013.png" /> are known as general Dirichlet series.
+
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 $-
 +
function]]). Series (1) with arbitrary exponents $  \lambda _ {n} $
 +
are known as general Dirichlet series.
  
 
==General Dirichlet series with positive exponents.==
 
==General Dirichlet series with positive exponents.==
Let, initially, the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032920/d03292014.png" /> be positive numbers. The analogue of the [[Abel theorem|Abel theorem]] for power series is then valid: If the series (1) converges at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032920/d03292015.png" />, it will converge in the half-plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032920/d03292016.png" />, and it will converge uniformly inside an arbitrary angle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032920/d03292017.png" />. The open domain of convergence of the series is some half-plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032920/d03292018.png" />. The number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032920/d03292019.png" /> is said to be the abscissa of convergence of the Dirichlet series; the straight line <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032920/d03292020.png" /> is said to be the axis of convergence of the series, and the half-plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032920/d03292021.png" /> 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, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032920/d03292022.png" />: The open domain in which the series converges absolutely (here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032920/d03292023.png" /> is the abscissa of absolute convergence). In general, the abscissas of convergence and of absolute convergence are different. But always:
+
Let, initially, the $  \lambda _ {n} $
 +
be positive numbers. The analogue of the [[Abel theorem|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:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032920/d03292024.png" /></td> </tr></table>
+
$$
 +
0  \leq  a - c  \leq  d ,\  \textrm{ where }  d = \overline{\lim\limits}\; _ {n\rightarrow \infty } \
  
and there exist Dirichlet series for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032920/d03292025.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032920/d03292026.png" />, the abscissa of convergence (abscissa of absolute convergence) is computed by the formula
+
\frac{ \mathop{\rm ln}  n }{\lambda _ {n} }
 +
,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032920/d03292027.png" /></td> </tr></table>
+
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
  
which is the analogue of the Cauchy–Hadamard formula. The case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032920/d03292028.png" /> is more complicated: If the magnitude
+
$$
 +
= = \overline{\lim\limits}\; _ {n \rightarrow \infty } 
 +
\frac{ \mathop{\rm ln}  | a _ {n} | }{\lambda _ {n} }
 +
,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032920/d03292029.png" /></td> </tr></table>
+
which is the analogue of the Cauchy–Hadamard formula. The case  $  d> 0 $
 +
is more complicated: If the magnitude
  
is positive, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032920/d03292030.png" />; if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032920/d03292031.png" /> and the series (1) diverges at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032920/d03292032.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032920/d03292033.png" />; if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032920/d03292034.png" /> and the series (1) converges at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032920/d03292035.png" />, then
+
$$
 +
\beta  = \overline{\lim\limits}\; _ {n \rightarrow \infty } 
 +
\frac{1}{\lambda _ {n} }
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032920/d03292036.png" /></td> </tr></table>
+
\mathop{\rm ln}  \left | \sum _ { i= } 1 ^ { n }  a _ {i} \right |
 +
$$
  
The sum of the series, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032920/d03292037.png" />, is an analytic function in the half-plane of convergence. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032920/d03292038.png" />, the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032920/d03292039.png" /> asymptotically behaves as the first term of the series, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032920/d03292040.png" /> (if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032920/d03292041.png" />). If the sum of the series is zero, then all coefficients of the series are zero. The maximal half-plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032920/d03292042.png" /> in which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032920/d03292043.png" /> is an analytic function is said to be the half-plane of holomorphy of the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032920/d03292044.png" />, the straight line <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032920/d03292045.png" /> is known as the axis of holomorphy and the number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032920/d03292046.png" /> is called the abscissa of holomorphy. The inequality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032920/d03292047.png" /> is true, and cases when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032920/d03292048.png" /> are possible. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032920/d03292049.png" /> be the greatest lower bound of the numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032920/d03292050.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032920/d03292051.png" /> is bounded in modulus in the half-plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032920/d03292052.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032920/d03292053.png" />). The formula
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032920/d03292054.png" /></td> </tr></table>
+
$$
 +
= \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
 
is valid, and entails the inequalities
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032920/d03292055.png" /></td> </tr></table>
+
$$
 +
| a _ {n} |  \leq 
 +
\frac{M ( \sigma ) }{e ^ {- \lambda _ {n} \sigma
 +
} }
 +
,\  M ( \sigma )  = \sup _ {- \infty < t < \infty }  | F (
 +
\sigma + it ) | ,
 +
$$
  
 
which are analogues of the Cauchy inequalities for the coefficients of a power series.
 
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032920/d03292056.png" />; it must, for example, tend to zero if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032920/d03292057.png" />. However, the following holds: Whatever the analytic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032920/d03292058.png" /> in the half-plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032920/d03292059.png" />, it is possible to find a Dirichlet series (1) such that its sum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032920/d03292060.png" /> will differ from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032920/d03292061.png" /> by an entire function.
+
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
 
If the sequence of exponents has a density
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032920/d03292062.png" /></td> </tr></table>
+
$$
 +
\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
 
the difference between the abscissa of convergence (the abscissas of convergence and of absolute convergence coincide) and the abscissa of holomorphy does not exceed
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032920/d03292063.png" /></td> </tr></table>
+
$$
 +
\delta  = \overline{\lim\limits}\; _ {n \rightarrow \infty } 
 +
\frac{1}{\lambda _ {n} }
  
and there exist series for which this difference equals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032920/d03292064.png" />. The value of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032920/d03292065.png" /> may be arbitrary in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032920/d03292066.png" />; in particular, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032920/d03292067.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032920/d03292068.png" /> then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032920/d03292069.png" />. The axis of holomorphy has the following property: On any of its segments of length <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032920/d03292070.png" /> the sum of the series has at least one singular point.
+
\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 ) ,
 +
$$
  
If the Dirichlet series (1) converges in the entire plane, its sum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032920/d03292071.png" /> is an entire function. Let
+
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.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032920/d03292072.png" /></td> </tr></table>
+
If the Dirichlet series (1) converges in the entire plane, its sum  $  F ( s) $
 +
is an entire function. Let
  
then the R-order of the entire function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032920/d03292073.png" /> (Ritt order) is the magnitude
+
$$
 +
\overline{\lim\limits}\; _ {n \rightarrow \infty } \
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032920/d03292074.png" /></td> </tr></table>
+
\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
 
Its expression in terms of the coefficients of the series is
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032920/d03292075.png" /></td> </tr></table>
+
$$
 +
-
 +
\frac{1} \rho
 +
  = \overline{\lim\limits}\; _ {n \rightarrow \infty } \
  
One can also introduce the concept of the R-type of a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032920/d03292076.png" />.
+
\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
 
If
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032920/d03292077.png" /></td> </tr></table>
+
$$
 +
\overline{\lim\limits}\; _ {n \rightarrow \infty } 
 +
\frac{n}{\lambda _ {n} }
 +
  = \
 +
\tau  < \infty
 +
$$
  
and if the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032920/d03292078.png" /> is bounded in modulus in a horizontal strip wider than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032920/d03292079.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032920/d03292080.png" /> (the analogue of one of the [[Liouville theorems|Liouville theorems]]).
+
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|Liouville theorems]]).
  
 
==Dirichlet series with complex exponents.==
 
==Dirichlet series with complex exponents.==
 
For a Dirichlet series
 
For a Dirichlet series
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032920/d03292081.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
$$ \tag{2 }
 +
F ( s)  = \sum _ {n = 1 } ^  \infty  a _ {n} e ^ {- \lambda _ {n} s }
 +
$$
  
with complex exponents <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032920/d03292082.png" />, the open domain of absolute convergence is convex. If
+
with complex exponents $  0 < | \lambda _ {1} | \leq  | \lambda _ {2} | \leq  \dots $,  
 +
the open domain of absolute convergence is convex. If
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032920/d03292083.png" /></td> </tr></table>
+
$$
 +
\lim\limits _ {n \rightarrow \infty } \
  
the open domains of convergence and absolute convergence coincide. The sum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032920/d03292084.png" /> of the series (2) is an analytic function in the domain of convergence. The domain of holomorphy of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032920/d03292085.png" /> is, generally speaking, wider than the domain of convergence of the Dirichlet series (2). If
+
\frac{ \mathop{\rm ln}  n }{\lambda _ {n} }
 +
  = 0 ,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032920/d03292086.png" /></td> </tr></table>
+
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.
 
then the domain of holomorphy is convex.
Line 93: Line 263:
 
Let
 
Let
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032920/d03292087.png" /></td> </tr></table>
+
$$
 +
\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|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} }
  
let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032920/d03292088.png" /> be an entire function of exponential type which has simple zeros at the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032920/d03292089.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032920/d03292090.png" />; let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032920/d03292091.png" /> be the Borel-associated function to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032920/d03292092.png" /> (cf. [[Borel transform|Borel transform]]); let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032920/d03292093.png" /> be the smallest closed convex set containing all the singular points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032920/d03292094.png" />, and let
+
e ^ {- \lambda t }  d \lambda ,\  n = 1 , 2 , . . . .
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032920/d03292095.png" /></td> </tr></table>
+
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 } \} $:
  
Then the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032920/d03292096.png" /> are regular outside <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032920/d03292097.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032920/d03292098.png" />, and they are bi-orthogonal to the system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032920/d03292099.png" />:
+
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032920/d032920100.png" /></td> </tr></table>
+
\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}
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032920/d032920101.png" /> is a closed contour encircling <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032920/d032920102.png" />. If the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032920/d032920103.png" /> are continuous up to the boundary of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032920/d032920104.png" />, the boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032920/d032920105.png" /> may be taken as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032920/d032920106.png" />. To an arbitrary analytic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032920/d032920107.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032920/d032920108.png" /> (the interior of the domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032920/d032920109.png" />) which is continuous in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032920/d032920110.png" /> one assigns a series:
+
\right .$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032920/d032920111.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>
+
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:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032920/d032920112.png" /></td> </tr></table>
+
$$ \tag{3 }
 +
F ( s)  \sim  \sum _ {n = 1 } ^  \infty 
 +
a _ {n} e ^ {\lambda _ {n} s } ,
 +
$$
  
For a given bounded convex domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032920/d032920113.png" /> it is possible to construct an entire function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032920/d032920114.png" /> with simple zeros <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032920/d032920115.png" /> such that for any function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032920/d032920116.png" /> analytic in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032920/d032920117.png" /> and continuous in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032920/d032920118.png" /> the series (3) converges uniformly inside <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032920/d032920119.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032920/d032920120.png" />. For an analytic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032920/d032920121.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032920/d032920122.png" /> (not necessarily continuous in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032920/d032920123.png" />) it is possible to find an entire function of exponential type zero,
+
$$
 +
a _ {n}  =
 +
\frac{1}{2 \pi i }
 +
\int\limits _ {\partial  D
 +
bar } F ( t) \psi _ {n} ( t)  d t ,\  n \geq  1 .
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032920/d032920124.png" /></td> </tr></table>
+
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,
  
and a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032920/d032920125.png" /> analytic in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032920/d032920126.png" /> and continuous in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032920/d032920127.png" />, such that
+
$$
 +
M ( \lambda )  = \sum _ {n = 0 } ^  \infty  c _ {n} \lambda  ^ {n} ,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032920/d032920128.png" /></td> </tr></table>
+
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
 
Then
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032920/d032920129.png" /></td> </tr></table>
+
$$
 +
\phi ( s)  = \sum _ {n = 0 } ^  \infty  a _ {n} M ( \lambda _ {n} )
 +
e ^ {\lambda _ {n} s } ,\  s \in D .
 +
$$
  
The representation of arbitrary analytic functions by Dirichlet series in a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032920/d032920130.png" /> was also established in cases when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032920/d032920131.png" /> is the entire plane or an infinite convex polygonal domain (bounded by a finite number of rectilinear segments).
+
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====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.F. Leont'ev,  "Exponential series" , Moscow  (1976)  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  S. Mandelbrojt,  "Dirichlet series, principles and methods" , Reidel  (1972)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.F. Leont'ev,  "Exponential series" , Moscow  (1976)  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  S. Mandelbrojt,  "Dirichlet series, principles and methods" , Reidel  (1972)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====

Revision as of 19:35, 5 June 2020


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:

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

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

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=37383
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