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 $$\sum_n a_n e^{-\lambda_nz}$$ 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 $1$ ) 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 $f(z)=\sum_n a_n e^{-\lambda_nz}$ be a Dirichlet series.

1. If $f$ converges at $z=z_0$ , then $f$ converges uniformly in the region $$\Re(z-z_0)\ge 0\qquad -\alpha\le\arg(z-z_0)\le \alpha$$ where $\alpha$ is any real number such that $0<\alpha<\pi/2$ . (Such a region is known as a ``Stoltz angle''.)
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. $f=0$ identically if and only if all the coefficients $a_n$ are zero.

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

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.