PlanetMath: Dirichlet series

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Dirichlet series

(Definition)

Let $(\lambda_n)_{n\ge 1}$ be an increasing sequence of positive real numbers tending to $\infty$ . A Dirichlet series with exponents $(\lambda_n)$ is a series of the form

$\displaystyle \sum_n a_n e^{-\lambda_nz}$

where

and all the

are complex numbers.

An ordinary Dirichlet series is one having $\lambda_n=\log n$ for all . It is written

$\displaystyle \sum\frac{a_n}{n^z}\;.$

The best-known examples are the Riemann zeta function (in which

is the constant

) and the more general Dirichlet L-series (in which the mapping $n\mapsto a_n$ is multiplicative and periodic).

When $\lambda_n=n$ , the Dirichlet series is just a power series in the variable $e^{-z}$ .

The following are the basic convergence properties of Dirichlet series. There is nothing profound about their proofs, which can be found in [1] and in various other works on complex analysis and analytic number theory.

Let $f(z)=\sum_n a_n e^{-\lambda_nz}$ be a Dirichlet series.

If converges at , then converges uniformly in the region
$\displaystyle \Re(z-z_0)\ge 0\qquad -\alpha\le\arg(z-z_0)\le \alpha$
where $\alpha$ is any real number such that $0<\alpha<\pi/2$ . (Such a region is known as a “Stoltz angle”.)
Therefore, if converges at , its sum defines a holomorphic function on the region $\Re(z)>\Re(z_0)$ , and moreover $f(z)\to f(z_0)$ as $z\to z_0$ within any Stoltz angle.
identically if and only if all the coefficients are zero.

So, if

converges somewhere but not everywhere in $\mathbb{C}$ , then the domain of its convergence is the region $\Re(z)>\rho$ for some real number $\rho$ , which is called the abscissa of convergence of the Dirichlet series. The abscissa of convergence of the series $f(z)=\sum_n \vert a_n\vert e^{-\lambda_nz}$ , if it exists, is called the abscissa of absolute convergence of

Now suppose that the coefficients are all real and nonnegative. If the series converges for $\Re(z)>\rho$ , and the resulting function admits an analytic extension to a neighbourhood of $\rho$ , then the series converges in a neighbourhood of $\rho$ . Consequently, the domain of convergence of (unless it is the whole of $\mathbb{C}$ ) is bounded by a singularity at a point on the real axis.

Finally, return to the general case of any complex numbers , but suppose $\lambda_n=\log n$ , so is an ordinary Dirichlet series $\sum\frac{a_n}{n^z}$ .

If the sequence is bounded, then converges absolutely in the region $\Re(z)>1$ .
If the partial sums $\sum_{n=k}^l a_n$ are bounded, then converges (not necessarily absolutely) in the region $\Re(z)>0$ .

References

1: Jean-Pierre Serre.
A Course in Arithmetic, chapter VI.
Springer-Verlag, 1973.
Zbl 0256.12001.
2: E. C. Titchmarsh.
The Theory of Functions.
Oxford Univ. Press, second edition, 1958.
Zbl 0336.30001.

"Dirichlet series" is owned by bbukh. [ full author list (2) | owner history (1) ]

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Also defines:

ordinary Dirichlet series, Stoltz angle, abscissa of convergence, abscissa of absolute convergence

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Cross-references: partial sums, converges absolutely, real axis, point, bounded, neighbourhood, function, domain, coefficients, holomorphic function, sum, region, converges uniformly, converges, analytic number theory, complex analysis, variable, power series, periodic, multiplicative, mapping, Dirichlet L-series, Riemann zeta function, complex numbers, series, exponents, real numbers, positive, sequence, increasing
There are 6 references to this entry.

This is version 5 of Dirichlet series, born on 2003-10-09, modified 2006-09-05.
Object id is 4764, canonical name is DirichletSeries.
Accessed 9583 times total.

Classification:

AMS MSC:

30B50 (Functions of a complex variable :: Series expansions :: Dirichlet series and other series expansions, exponential series)

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