Dirichlet series

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

$$\sum _n{a}_n{e}^{-{\lambda }_{n}z}$$

where $z$ and all the $a_n$ are complex numbers.

An ordinary Dirichlet series is one having ${\lambda }_n=\log n$ for all $n$. It is written

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

The best-known examples are the Riemann zeta function (in which $a_n$ is the constant (http://planetmath.org/ConstantFunction) $1$) 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 $f(z)={\sum }_n{a}_n{e}^{-{\lambda }_{n}z}$ be a Dirichlet series.

1. If $f$ converges at $z=z_0$, then $f$ converges uniformly in the region

   $$\mathrm{\Re }(z-z_0)\ge 0\mathit{\hspace{1em}\hspace{1em}}-\alpha \le \arg (z-z_0)\le \alpha$$

   where $\alpha$ is any real number such that . (Such a region is known as a “Stoltz angle”.)

2. Therefore, if $f$ converges at $z_0$, its sum defines a holomorphic function on the region $\mathrm{\Re }(z)>\mathrm{\Re }(z_0)$, and moreover $f(z)\to f(z_0)$ as $z\to z_0$ within any Stoltz angle.

3. $f=0$ identically if and only if all the coefficients $a_n$ are zero.

So, if $f$ converges somewhere but not everywhere in $ℂ$, then the domain of its convergence is the region $\mathrm{\Re }(z)>\rho$ for some real number $\rho$, which is called the abscissa of convergence of the Dirichlet series. The abscissa of convergence of the series $f(z)={\sum }_n|a_n|e^{-{\lambda }_{n}z}$, if it exists, is called the abscissa of absolute convergence of $f$.

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

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

1. If the sequence $(a_n)$ is bounded, then $f$ converges absolutely in the region $\mathrm{\Re }(z)>1$.

2. If the partial sums ${\sum }_{n=k}^l{a}_n$ are bounded, then $f$ converges (not necessarily absolutely) in the region $\mathrm{\Re }(z)>0$.

Title	Dirichlet series
Canonical name	DirichletSeries
Date of creation	2013-03-22 13:59:22
Last modified on	2013-03-22 13:59:22
Owner	bbukh (348)
Last modified by	bbukh (348)
Numerical id	8
Author	bbukh (348)
Entry type	Definition
Classification	msc 30B50
Related topic	DirichletLFunction
Related topic	RiemannZetaFunction
Related topic	DirichletLSeries
Defines	ordinary Dirichlet series
Defines	Stoltz angle
Defines	abscissa of convergence
Defines	abscissa of absolute convergence