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edited Jul 9, 2010 at 11:48

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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 both cases ofDirichlet series in bounded domains and some unbounded convex domains (including half-planes). The problem is that his results may not be directly applicable to the `ordinary' Dirichlet series you are interested inwith $\lambda_n=\ln n$. A A typical condition on the sequence $\lambda_n$ which appears in Leont'ev's results in the case oftheorem for half-planes is $$\lim\limits_{n\to\infty}\frac{n}{\lambda_n^\rho}=\sigma,\quad 0<\sigma<\infty$$ with some as follows $\rho>1$, which(link to the original article in Russian).

Theorem. For every $\rho>1$, there is a sequence $\lambda_n>0$, $n\in\mathbb N$, satisfying the condition $$\lim\limits_{n\to\infty}\frac{n}{\lambda_n^\rho}=\tau,\quad 0<\tau<\infty,$$ such that any function $f$, which is analytic in the right half-plane $\Re z > 0$ , can be represented in the form $$f(z)=\sum\limits_{n=1}^{\infty}a_n e^{-\lambda_n z}+\Phi(z),\qquad \Re z > 0,$$ where $\Phi$ is entire.

This obviously fails whendoesn'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~~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).~~ (PDF files)Edit: sent.) By the way, the papers you've mentioned both deal with the case of convergence in a bounded domain $D$.

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 both cases of bounded domains and some unbounded convex domains (including half-planes). The problem is that his results may not be directly applicable to the `ordinary' Dirichlet series you are interested in. A typical condition on the sequence $\lambda_n$ which appears in Leont'ev's results in the case of half-planes is $$\lim\limits_{n\to\infty}\frac{n}{\lambda_n^\rho}=\sigma,\quad 0<\sigma<\infty$$ with some $\rho>1$, which obviously fails when $\lambda_n=\ln n$.

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). By the way, the papers you've mentioned both deal with the case of convergence in a bounded domain $D$.

Theorem. For every $\rho>1$, there is a sequence $\lambda_n>0$, $n\in\mathbb N$, satisfying the condition $$\lim\limits_{n\to\infty}\frac{n}{\lambda_n^\rho}=\tau,\quad 0<\tau<\infty,$$ such that any function $f$, which is analytic in the right half-plane $\Re z > 0$ , can be represented in the form $$f(z)=\sum\limits_{n=1}^{\infty}a_n e^{-\lambda_n z}+\Phi(z),\qquad \Re z > 0,$$ where $\Phi$ is entire.

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 $D$.

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created Jul 8, 2010 at 16:39

Andrey Rekalo

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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 both cases of bounded domains and some unbounded convex domains (including half-planes). The problem is that his results may not be directly applicable to the `ordinary' Dirichlet series you are interested in. A typical condition on the sequence $\lambda_n$ which appears in Leont'ev's results in the case of half-planes is $$\lim\limits_{n\to\infty}\frac{n}{\lambda_n^\rho}=\sigma,\quad 0<\sigma<\infty$$ with some $\rho>1$, which obviously fails when $\lambda_n=\ln n$.