The number of restricted solutions of some systems of linear congruences (Q1231927)

Formulas for the number of solutions of the system of congruences \(n_i\equiv x_{i1} +\ldots+x_{is} \pmod r\), \(i =1, \ldots, t\), are obtained when the solutions are required to satisfy certain conditions. For example if \(D\) is a nonempty set of divisors of \(r\) and the solutions are required to satisfy \(1\le x_{ij}\le r\) and \((x_{1j},x_{2j},\ldots,x_{tj}, r)\in D\) for \(j = 1, \ldots,s\), the number of such solutions is found as a special case of a general method which uses complex exponential sums, and even and totally even functions introduced by E. Cohen.