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

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

The best-known examples are the Riemann zeta function (in which $a_{n}$ is the constant $1$) 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. 1.

   If $f$ converges at $z=z_{0}$, then $f$ converges uniformly in the region

   $$\Re(z-z_{0})\geq 0\qquad-\alpha\leq\arg(z-z_{0})\leq\alpha$$

   where $\alpha$ is any real number such that $0<\alpha<\pi/2$. (Such a region is known as a “Stoltz angle”.)

2. 2.

   Therefore, if $f$ converges at $z_{0}$, 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.

3. 3.

   $f=0$ identically if and only if all the coefficients $a_{n}$ are zero.

So, if $f$ 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 $f(z)=\sum_{n}|a_{n}|e^{{-\lambda_{n}z}}$, 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 $\Re(z)>\rho$, and the resulting function admits an analytic extension 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 $\mathbb{C}$) 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. 1.

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

2. 2.

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