Let $F(s)=\sum_{n\geq 1}\frac{a_n}{n^s}$ be a Dirichlet series with (finite) abscissa of absolute convergence $\sigma_a$. It can be shown that $\forall \sigma >\sigma_a:$
$$\lim_{T\to\infty}\frac{1}{2T}\int_{-T}^{T}F(\sigma+ it)n^{it}\mathrm{d}t=\frac{a_n}{n^{\sigma}}.$$
The natural question arises, given some function $F$ holomorphic in some half-plane, under what conditions does it have a representation as a Dirichlet series. I believe this is a very broad question, so I would actually like to make things a bit more specific.
For fixed $c\geq 0$ let $H:=H_c:=\{z\in\mathbb{C}:\Re(z)>c\}$ be some half-plane and let $f\in\mathcal{O}(H)$. For $\sigma> c$ define the linear functional
$$\Phi_{n,\sigma}: f\mapsto\lim_{T\to\infty}\frac{1}{2T}\int_{-T}^{T}f(\sigma + it)n^{it}\mathrm{d}t$$
I have intentionally left out the actual domain of $\Phi_{n,\sigma}$ in $\mathcal{O}(H)$ (since it is rather part of the general question than a known fact). It can be easily seen though that $\Phi_{n,\sigma}$ is not well-defined on the whole $\mathcal{O}(H)$. Let $\sigma_0>c$ be fixed real number.
(Q1) Provided $\{\Phi_{n,\sigma_0}(f)\} _{n\in\mathbb{N}}$ exists, does it follow that $\{ \Phi_{n,\sigma}(f)\} _{n\in\mathbb{N}}$ exists for all $\sigma>\sigma_0$?
(Q2) Provided $\{\Phi_{n,\sigma_0}(f)\} _{n\in I}$ exists, where $I\subset\mathbb{N}$ is some infinite subset, does it follow that $\Phi_{n,\sigma_0}(f)$ exists for all $n\in\mathbb{N}$? How about sufficiently large finite subset $I\subset\mathbb{C}$?
(Q3) Provided that $n^{\sigma_0}\Phi_{n,\sigma_0}(f)=:a_n$ exists for all $n\in\mathbb{N}$. Does it follow that the Dirichlet series $\sum_{n\geq 1}\frac{a_n}{n^s}$ is absolute(?) convergent in some half-plane? If it is convergent, does it represent $f$ in that half-plane?
(Q4) And the more general question: Are there any known conditions when an analytic function admits expansion as an ordinary Dirichlet series?
I am also aware of the existence of a series of papers of A.F. Leont'ev on the representations of analytic functions as Dirichlet series, e.g. "On the representation of analytic functions by Dirichlet series", A. F. Leont'ev 1969 Math. USSR Sb. 9 111 and "On conditions of expandibility of analytic functions in Dirichlet series", A. F. Leont'ev 1972 Math. USSR Izv. 6 1265, etc. English translations as well as some of the original are available at iopsciences. Unfortunately for me, I don´t have institutional access to those 🙁
However, while Leontev´s papers appear to be fundamental for the subject, they all date back to the period 1969-1975. So I was hoping that there might be some good serveys or other types of good references summarizing the recent developments, respectively the most general results in the subject so far and that would be also "easier to have" than the aforementioned papers. Also, per Andrey Rekalo´s comment it seems that Leont'ev´s work is not really applicable to the more specific case of representation by ordinary Dirichlet series.
Thank you in advance for any input!
Best Answer
A.F. Leont'ev continued to work on general Dirichlet series well into 1980s (until his death in 1987). Actually, he published three monographs on the subject from 1976 to 1983! He made a short summary of his earlier results for the 1974 ICM in Vancouver (a free preview of the lecture is available here).
A.F. Leont'ev obtained in some sense final results on the representation of analytic functions by general Dirichlet series of the form $$f(s)=\sum\limits_{n=1}^{\infty}a_n e^{-\lambda_n s},\quad s\in D\subset \mathbb C.$$ He studied Dirichlet series in bounded and unbounded convex domains (including half-planes). The problem is that his results may not be directly applicable to the `ordinary' Dirichlet series with $\lambda_n=\ln n$. A typical Leont'ev's theorem for half-planes is as follows (link to the original article in Russian).
This obviously doesn't cover the case $\lambda_n=\ln n$, $n\in\mathbb N$.
Anyway, if you're interested and if you have a colleague who speaks Russian I can send you a couple of original articles by Leont'ev (PDF files).(Edit: sent.) By the way, the papers you've mentioned both deal with the case of convergence in a bounded domain $D$.