Dirichlet series

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:

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

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

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

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)
[a1]	G.H. Hardy, M. Riesz, "The general theory of Dirichlet series" , Cambridge Univ. Press (1915) Zbl 45.0387.03