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# Dirichlet series

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

$\sum_{n}a_{n}e^{{-\lambda_{n}z}}$ |

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 $1$) 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

$\Re(z-z_{0})\geq 0\qquad-\alpha\leq\arg(z-z_{0})\leq\alpha$ where $\alpha$ is any real number such that $0<\alpha<\pi/2$. (Such a region is known as a “Stoltz angle”.)

2. 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.

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. 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.

## Mathematics Subject Classification

30B50*no label found*

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