The hexagonal versus the square lattice
Speaker: Pieter Moree (Max-Planck-Institute for Mathematics, Bonn)
Date: Wed 2nd November 2005
Location: Mathematical Sciences Seminar Room
Schmutz-Schaller formulated in 1995 a conjecture concerning lattices of dimensions 2 to 8 and proved its analogue in hyperbolic geometry. As a particular case he mentioned that the hexagonal lattice ought to be 'better' than the square lattice. This statement is equivalent with the statement that for every x the number of integers n<=x that can be written as a sum of two squares is not less than the number of integers m<=x that can be written as a sum of square and three times a square.
Together with Herman te Riele (CWI, Amsterdam) I recently proved this by methods from computational number theory and the asymptotic theory of arithmetic functions.
As a byproduct I disproved some claims on the divisibility of the tau-function Ramanujan made in his unpublished intriguing manuscript on the partition and tau function (two famous functions in number theory).
(This talk is part of the K-Theory, Quadratic Forms and Number Theory series.)