Nonvanishing theorems for L-functions of modular forms and their derivatives

Inventiones mathematicae -     

LEED rotation diagrams for Al

We present a comparison between the experimental and theoretical LEED intensities for the (001) surface of Al. The intensity curves were calculated for constant energy and the angle θ as a function of the azimuthai angle ? (“rotation diagrams”) using the band structure-matching approach. The band structure was obtained using an OPW-based Pseudopotential with one adjustable parameter (the core shift). The experimental curves were obtained with a new goniometer which allows an angular resolution of 0.1°. The experimental and calculated (elastic and inelastic) spectra are compared and discussed.     

Multiple Dirichlet Series and Moments of Zeta and L-Functions

This paper develops an analytic theory of Dirichlet series in several complex variables which possess sufficiently many functional equations. In the first two sections it is shown how straightforward conjectures about the meromorphic continuation and polar divisors of certain such series imply, as a consequence, precise asymptotics (previously conjectured via random matrix theory) for moments of zeta functions and quadratic L-series. As an application of the theory, in a third section, we obtain the current best known error term for mean values of cubes of cent ral values of Dirichlet L-series. The methods utilized to derive this result are the convexity principle for functions of several complex-variables combined with a knowledge of groups of functional equations for certain multiple Dirichlet series.     

Some analytic bounds for zeta functions and class numbers

Inventiones mathematicae -     

Perturbation theories in low energy electron diffraction: t-matrix and τ-matrix approaches

Two perturbation approaches to low energy electron diffraction (LEED) are discussed within the inelastic collision model: t-matrix approach recently proposed by Tong, Rhodin and Tait (all scatterings treated to third order), and τ-matrix approach (layer scattering is treated exactly while all interlayer scatterings are treated to third order). Both approaches are applied to the calculation of rotation diagrams for the (001) surface of Al. In the case of a single layer third-order perturbation results for the magnitude and phase of the scattering amplitude are compared to the exact results and found to be in fairly good agreement over a large range of elastic scattering cross-sections. When all scatterings (up to third order) are considered for an infinite crystal, the differences between the exact and third-order t-matrix perturbation resutls become quite substantial ; marked improvement is achieved if the layer scattering is treated exactly, but the remaining difference may be significant enough (depending on the application) to suggest that interlayer scattering may not always be treated by perturbation, and thus renormalization of the type proposed by Pendry may be necessary.     

Determining modular forms on $${S L_2(\mathbb{Z})}$$ by central values of convolution L-functions

We show that a cuspidal normalized Hecke eigenform g of level one and even weight is uniquely determined by the central values of the family of Rankin– Selberg L-functions \({L(s, f\otimes g)}\) , where f runs over the Hecke basis of the space of cusp forms of level one and weight k with k varying over an infinite set of even integers.