Abstract
Our object is to obtain formulae for the number of partitions of the vector (n[l],...,n[j]) into parts whose components lie in certain congruence classes (modulo M), where M, n[l],...,n[j] are all positive integers. The asymptotic expression obtained for the generating function of multi-partite partitions enables us to consider all large n[l],...,n[j] each of approximately the same order of magnitude. If M = p (a prime) partitions always exist. For composite values of M non-zero partitions always exist if the Vector (n[l],...,n[j])is "parallel" to a certain j-dimensional "plane" which is determined by the residues (modulo M) of the problem. This can be interpreted geometrically when j=3. We find, even for large values of n[l],...,n[j], that the chance of a non-zero partition of (n[l],...,n[j]) occurring is very small and in the asymptotic formulae obtained this is caused by the strong thinning effect of a certain exponential sum which is zero except when each n(1 < 1 < j) lies in a certain residue class (mod M). Certain special cases of these formulae are considered for j=2 . In order to estimate the number of partitions in these formulae the evaluation of a certain integral I (z[l],...,z[j]) is required. This is considered in the last six sections where asymptotic expansions are obtained for I (z[l],...,z[j]) when every z is small and approximate expressions are given for I (z[l],...,z[j]) when every z is real. Finally, exact formulae are given when every z is real and rational.