The number of lattice rules

Abstract

A lattice rule is a quadrature rule for integration over ans-dimensional hypercube that employsN abscissas located on a lattice, chosen to conform to certain specifications. In this paper we determine the numberv _s(N) of distinctN-points-dimensional lattice rules. We show that, in general,

$$v_s (MN) \leqslant v_s (M)v_s (N),$$

equality being attained if and only ifM andN are mutually prime. This is used to establish that, whenL has prime factor expansion$\prod\limits_j {p_j^{t_J } ,}$, then$v_s (L) = \prod\limits_j {v_s (p_j^{t_J } )'}$ where

$$v_s (p^t ) = \prod\limits_{i = 1}^{s - 1} {[(p^{i + t} - 1)/(p^i - 1)]; s \geqslant 1,t \geqslant 0.} $$