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For fixed  n  and  e , the number of subgroups of indexinis polynomial in  p . Is this true for subrings inof index? Equivalently, is the subring zeta function uniform? In this paper, we make progress toward answering these questions. First, we connect counting subrings of indexinto counting solutions to equations modulo various powers of  p . We show some subsets of these equations have a polynomial number of solutions, but others do not. Finally, we give an explicit polynomial formula for the number of ‘irreducible’ subrings of indexin.

Is the number of subrings of index p e in ℤ n polynomial in p ?