We consider various Hilbert spaces of Dirichlet series whose norms are given by weighted l2 norms of the Dirichlet coefficients. We characterize the multiplier algebras for some of these spaces. 0 Introduction Let w = {wn}n=n0 be a sequence of positive numbers. In this paper we are concerned with Hilbert spaces of functions representable by Dirichlet series: H w = {

1 The classical Möbius function 1.1 Arithmetic functions and Dirichlet series. Write N (meant to encode some arithmetic feature of each number n). To each arithmetic function f one associates a Dirichlet series F (s) = n≥1 f (n) n s thought of as a function defined on some open set of the complex plane. Analytic number theory is to a large extent the study of arithmetic functions in terms of th...

We prove some conditions on the existence of natural boundaries of Dirichlet series. We show that generically the presumed boundary is the natural one. We also give an application of natural boundaries in determining asymptotic results.

We derive a general expression for the Hankel determinants of a Dirichlet series F (s) and derive the asymptotic behavior for the special case that F (s) is the Riemann zeta function. In this case the Hankel determinant is a discrete analogue of the Selberg integral and can be viewed as a matrix integral with discrete measure. We brie y comment on its relation to Plancherel measures.

An L-function, as the term is generally understood, is a Dirichlet series in one complex variable s with analyic continuation to all complex s, a functional equation and an Euler product. The coefficients are thus multiplicative. Weyl group multiple Dirichlet series are a new class of Dirichlet series that differ from L-functions in two ways. First, they are Dirichlet series in several complex ...

satisfying zZ\ \^A < °°, if and only if f is) is bounded away from zero in the half-plane 0-3:0. This discovery amounts to a determination of the spectrum of the Banach-algebra element associated with fis), and it thus makes available for the theory of ordinary Dirichlet series the well-known theorem of Gelfand on analytic functions of Banach-algebra elements (see §24D of [7]) and its generaliz...

Let Φ be a reduced root system of rank r. A Weyl group multiple Dirichlet series for Φ is a Dirichlet series in r complex variables s1, . . . , sr, initially converging for Re(si) sufficiently large, that has meromorphic continuation to C and satisfies functional equations under the transformations of C corresponding to the Weyl group of Φ. A heuristic definition of such series was given in [2]...

We provide a general theorem for evaluating trigonometric Dirichlet series of the form ∑ n>1 f(πnτ) ns , where f is an arbitrary product of the elementary trigonometric functions, τ a real quadratic irrationality and s an integer of the appropriate parity. This unifies a number of evaluations considered by many authors, including Lerch, Ramanujan and Berndt. Our approach is based on relating th...