Dirichlet series

Let ${({\lambda }_n)}_{n\ge 1}$ be an increasing sequence ofpositive real numbers tending to $\mathrm{\infty }$.A Dirichlet series with exponents $({\lambda }_n)$ isa 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 seriesin the variable $e^{-z}$.

The following are the basic convergence properties of Dirichlet series.There is nothing profound about their proofs, which can be foundin [1] and in various other works on complex analysis and analyticnumber 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 holomorphicfunction 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 $ℂ$, thenthe domain of its convergence is the region $\mathrm{\Re }(z)>\rho$ forsome real number $\rho$, which is called the abscissa of convergenceof 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 functionadmits 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 wholeof $ℂ$) is bounded by a singularity at a point on the real axis.

Finally, return to the general case of any complex numbers $(a_n)$, butsuppose ${\lambda }_n=\log n$, 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 theregion $\mathrm{\Re }(z)>1$.

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