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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|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|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.==
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:

$$ 
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|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|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====
<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====

====References====
<table>
<TR><TD valign="top">[a1]</TD> <TD valign="top">  G.H. Hardy,   M. Riesz,   "The general theory of Dirichlet series" , Cambridge Univ. Press  (1915) {{ZBL|45.0387.03}}</TD></TR>
</table>