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]...

Every Jacobi cusp form of weight k and index m over SL2(Z) Z 2 is in correspondence with 2m Dirichlet series constructed with its Fourier coefficients. The standard way to get from one to the other is by a variation of the Mellin transform. In this paper, we introduce a set of integral kernels which yield the 2m Dirichlet series via the Petersson inner product. We show that those kernels are Ja...

We are presenting in this paper a detailed account of the methods used to compute the three-dimensional two-point galaxy correlation function in the VIMOS-VLT deep survey (VVDS). We investigate how instrumental selection effects and observational biases affect the measurements and identify the methods to correct them. We quantify the accuracy of our correction method using an ensemble of fifty ...

For entire functions, the notions of their growth indicators such as Ritt order are classical in complex analysis. But the concepts of relative Ritt order of entire functions and as well as their technical advantages of not comparing with the growths of $exp exp z$ are not at all known to the researchers of this area. Therefore the studies of the growths of entire functions in the light of thei...