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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!

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?

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, in particular 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. But, unfortunately etc. English translations as well as some of the original are available at iopsciences. Unfortunately for me, I don´t have any 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 developmentsresp. , 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!

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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 $\Phi_{n,\sigma_0}(f)=:a_n$ 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}$is absolute(?) convergent in some (proper) half-plane? If it is convergent, does this Dirichlet series it represent$f\$ in that half-plane?

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, in particular "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. But, unfortunately for me, I don´t have any 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 resp. the most general results in the subject so far and that would be also "easier" easier to have" than the aforementioned papers.

Thank you in advance for any input!

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