Which functions do Dirichlet series represent?

Question

I'm reading Serre's $\textit{A course in Arithmetic}$ where he defines a Dirichlet series to be an infinite sum of the form $$f(z) = \sum\limits_{n=1}^{\infty} a_ne^{-\lambda_nz} $$ where $\lambda_n$ is an increasing sequence of reals diverging to infinity and $a_n \in \mathbb{C}$.

One can associate with these series, half-planes $H$ (including $\mathbb{C}$ and $\varnothing$) on which they converge. More precisely, if $f$ converges at $z_0$, then it must converge uniformly on compact subsets of the half plane $\Re(z)>\Re(z_0)$. This shows that $f$ is holomorphic here too.

Given a holomorphic $f$ on some half plane $H$, is it representable by a Dirichlet series?

Going through the basic theorems hasn't thrown up any obvious holomorphic functions precluded from having such a representation. Am I missing something?

Answer 1

The idea is that if the Dirichlet series converges at some $z_0$ then $$\frac{f(z+z_0)}{z} =\sum_{n=1}^\infty a_n e^{-\lambda_n z_0}\frac{e^{-\lambda_n z}}{z}= \sum_{n=1}^\infty a_n e^{-\lambda_n z_0}\int_{\lambda_n}^\infty e^{-tz}dt = \int_{\lambda_0}^\infty (\sum_{\lambda_n \le t} a_n e^{-\lambda_n z_0}) e^{-tz}dt$$

Thus for $\Re(z) > 0$,

$\frac{f(z+z_0)}{z}$ is the Laplace transform of the piecewise constant bounded function $\sum_{\lambda_n \le t} a_n e^{-\lambda_n z_0}$ supported on $t\ge \lambda_0$.

In particular $\frac{f(z)}{z}$ is $L^2$ on vertical lines and it decays uniformly as $|\Im(z)|\to \infty$.

Conversely if for $\Re(z) >\Re(z_0)$, $\frac{f(z)}{z}$ is $L^2$ on vertical lines and it decays uniformly as $|\Im(z)|\to \infty$ then the inverse Fourier/Laplace transform $$F(t)=\mathcal{L}^{-1}[\frac{f(z+z_0)}{z}]$$ is well-defined (in $L^2$ sense) and it suffices to check if : it is piecewise constant and supported on $t \ge T$ to find if for some $a_n$ and some reals $\lambda_n<\lambda_{n+1}\to\infty$ $$F(t) = \sum_{\lambda_n \le t} a_n e^{-\lambda_n z_0}, \qquad f(z+z_0) = z\int_{\lambda_0}^\infty F(t)e^{-zt}dt= \sum_{n=1}^\infty a_n e^{-\lambda_n (z+z_0)}$$

Answer 2

Let $d(z) = \sum\limits_{n} a_n e^{-\lambda_n z}$ be a Dirichlet series converging in some non-empty half plane $H$. Proposition $6$ on page $66$ of the book mentioned implies that the sum must converge to $d(z)$ uniformly on the real line intersected with $H$. You can see this uniform convergence by using Abel's summation lemma.

I claim that the function $f(z)=z$ cannot be represented by a Dirichlet series on any non-empty $H$. If it were so, one could subtract the terms having $\lambda_n<0\ ^*$ to get a function which was bounded on the half real axis. This is impossible for a function of the form $$z - (a_1e^{-\lambda_1 z} + \dots + a_me^{-\lambda_m z}) $$ with $\lambda_1 < \dots < \lambda_m < 0$.

Additionally, as Conrad points out in the comments, Dirichlet series enjoy some 'almost-periodic' properties on vertical lines not seen for general holomorphic functions (further explanation would be nice at some point).

$^*$ The book actually assumes each $\lambda_n\geq 0$, so strictly speaking we do not have to deal with this case.