Introduction This paper tries to shed some light on the very classical topic of overconvergence for Dirichlet series, by employing results in the theory of innite order dier-ential operators with constant coecients ([3], [13]). The possibility of linking innite order dierential operators with gap theorems and related subjects such as overconvergence phenomena was rst suggested by Ehrenpreis in ...

Given a root system Φ of rank r and a global field F containing the n-th roots of unity, it is possible to define a Weyl group multiple Dirichlet series whose coefficients are n-th order Gauss sums. It is a function of r complex variables, and it has meromorphic continuation to all of C, with functional equations forming a group isomorphic to the Weyl group of Φ. Weyl group multiple Dirichlet s...

Associated to a newform f (z) is a Dirichlet series L f (s) with functional equation and Euler product. Hecke showed that if the Dirichlet series F (s) has a functional equation of the appropriate form, then F (s) = L f (s) for some holomorphic newform f (z) on Γ(1). Weil extended this result to Γ 0 (N) under an assumption on the twists of F (s) by Dirichlet characters. We show that, at least f...

Weyl group multiple Dirichlet series were associated with a root system Φ and a number field F containing the n-th roots of unity by Brubaker, Bump, Chinta, Friedberg and Hoffstein [3] and Brubaker, Bump and Friedberg [4] provided n is sufficiently large; their coefficients involve n-th order Gauss sums and reflect the combinatorics of the root system. Conjecturally, these functions coincide wi...

If F is a local field containing the group μn of n-th roots of unity, and if G is a split semisimple simply connected algebraic group, then Matsumoto [27] defined an n-fold covering group of G(F ), that is, a central extension of G(F ) by μn. Similarly if F is a global field with adele ring AF containing μn there is a cover G̃(AF ) of G(AF ) that splits over G(F ). The construction is built on i...

MIXED ZETA FUNCTIONS Pieter Mostert Ted Chinburg We examine Dirichlet series which combine the data of a distance function, u, a homogeneous degree zero function, φ, and a multivariable Dirichlet series, K. By using an integral representation and Cauchy’s residue formula, we show that under certain conditions on K, such functions extend to meromorphic functions on C, or to some region strictly ...

We introduce a concept called good oscillation. A function is called good oscillation, if its m-tuple integrals are bounded by functions having mild orders. We prove that if the error terms coming from summatory functions of arithmetical functions are good oscillation, then the Dirichlet series associated with those arithmetical functions can be continued analytically over the whole plane. We a...

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 [BB...

We consider the Dirichlet series ∞ ∑ k=2 akk −1−x =: f(x), x > 0, with coefficients ak ≥ 0 for all k. Among others, we prove exact estimates of certain weighted Lp-norms of f on the unit interval (0, 1) for any 0 < p <∞, in terms of the coefficients ak . Our estimation is based on the close relationship between Dirichlet series and power series. This enables us to derive exact estimates for int...

A general initial-boundary value problem of one-dimensional transient wave propagation in a multi-layered elastic medium due to arbitrary boundary or interface excitations (either prescribed tractions or displacements) is considered. Laplace transformation technique is utilised and the Laplace transform inversion is facilitated via an unconventional method, where the expansion of complex-valued...