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The sphere packing problem in dimension 24
Building on Viazovska's recent solution of the sphere packing problem in eight dimensions, we prove that the Leech lattice is the densest packing of congruent spheres in twenty-four dimensions and
Eisenstein series for higher-rank groups and string theory amplitudes
Scattering amplitudes of superstring theory are strongly constrained by the requirement that they be invariant under dualities generated by discrete subgroups, E_n(Z), of simply-laced Lie groups in
SL(2,Z)-invariance and D-instanton contributions to the $D^6 R^4$ interaction
The modular invariant coefficient of the $D^6R^4$ interaction in the low energy expansion of type~IIB string theory has been conjectured to be a solution of an inhomogeneous Laplace eigenvalue
Cancellation in additively twisted sums on GL(n)
<abstract abstract-type="TeX"><p>In a previous paper with Schmid we considered the regularity of automorphic distributions for <i>GL</i>(2,R), and its connections to other topics in number theory and
Automorphic distributions, L-functions, and Voronoi summation for GL(3)
This paper is third in a series of three, following "Summation Formulas, from Poisson and Voronoi to the Present" (math.NT/0304187) and "Distributions and Analytic Continuation of Dirichlet Series"
Riemann's zeta function and beyond
In recent years L-functions and their analytic properties have assumed a central role in number theory and automorphic forms. In this expository article, we describe the two major methods for proving
A general Voronoi summation formula for GL(n,Z)
In an earlier paper we derived an analogue of the classical Voronoi summation formula for automorphic forms on GL(3), by using the theory of automorphic distributions. The purpose of the present
Universal optimality of the $E_8$ and Leech lattices and interpolation formulas
We prove that the $E_8$ root lattice and the Leech lattice are universally optimal among point configurations in Euclidean spaces of dimensions $8$ and $24$, respectively. In other words, they