Dirichlet series and convolution equations (Q1118060)

There are two main methods to study analytic continuation properties of Dirichlet series \[ f(s)=\sum_{n\geq 1}a_ ne^{\lambda_ ns}\quad with\quad finite\quad density, \] i.e., \(n/\lambda_ n=0(1)\). One is due to Hadamard and consists of the use of entire functions \(\gamma\) that interpolate the values \(a_ n\) at the points \(z=\lambda_ n\) (cf. \textit{V. Bernstein}, Dirichlet series, Gauthier-Villars (1932)). The other is due to S. Mandelbrojt (Hermann (1936), Gauthier-Villars (1952)), it can be interpreted as that of considering the sequence \(\{a_ n\}\) as an element of the dual of a quotient space of a space of holomorphic functions. This is just another way of saying that f satisfies a convolution equation. We use this latter point of view to obtain theorems about Dirichlet series of \(\infty\) density, like \(\lambda_ n\sim \sqrt{n}\). Further work in this direction is in progress by the authors, relating singularity properties of Dirichlet series to vanishing theorems of cohomology. The case of \(\lambda_ n=\log n\) appear in the Ph.D thesis of R. Heymann. Our work is also related to recent work of Dzhrbashyan and of Arakelyan.