$$L$$ L -functions as distributions

Zero Spacing Distributions for Differenced L-Functions

The paper studies the local zero spacings of deformations of the Riemann ξ-function under certain averaging and differencing operations. For real h we consider the entire functions Ah(s) := 1 2 (ξ(s + h) + ξ(s− h)) and Bh(s) = 1 2i (ξ(s+ h)− ξ(s− h)) . For |h| ≥ 1 2 the zeros of Ah(s) and Bh(s) all lie on the critical line R(s) = 1 2 and are simple zeros. The number of zeros of these functions ...

Automorphic Distributions, L-functions, and Voronoi Summation for GL(3)

Automorphic distributions , L - functions , and Voronoi summation for GL

for the error term in Gauss’ classical circle problem, improving greatly on Gauss’ own bound O(x1/2). Though Voronoi originally deduced his formulas from Poisson summation in R2, applied to appropriately chosen test functions, one nowadays views his formulas as identities involving the Fourier coefficients of modular forms on GL(2), i.e., modular forms on the complex upper half plane. A discuss...

L–distributions on symmetric spaces

The notion of Lp–distributions is introduced on Riemannian symmetric spaces of noncompact type and their main properties are established. We use a geometric description for the topology of the space of test functions in terms of the Laplace–Beltrami operator. The techniques are based on a-priori estimates for elliptic operators. We show that structure theorems, similar to Rn, hold on symmetric ...

INVARIANT DISTRIBUTIONS ON l-SPACES

In a paper published by Ranga Rao [4] in the Annals of Mathematics, it is shown that the invariant measure carried by a nilpotent orbital can be extended to an invariant distribution on the ambient Lie algebra. This paper provides an alternative proof by a method proposed by Gordon Savin.