Lattice rules are integration rules for approximating integrals of periodic functions over the s-dimensional unit cube. One criterion for measuring the 'goodness' of lattice rules is the quantity R . This quantity is defined as a sum which contains about Ns~l terms, where TV is the number of quadrature points. Although various bounds involving R are known, a procedure for calculating R itself does not appear to have been given previously. Here we show how an asymptotic series can be used to

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... in an accurate approximation to R . Whereas an efficient direct calculation of R requires OiNnx) operations, where nx is the largest 'invariant' of the rule, the use of this asymptotic expansion allows the operation count to be reduced to OiN). A complete error analysis for the asymptotic expansion is given. The results of some calculations of R are also given.