Lower bounds for the number of subrings in $\mathbb{Z}^n$

Abstract

Let fn(k) be the number of subrings of index k in Z n . We prove two new lower bounds for fn(p e ) when e ≥ n − 1. Using these bounds, we study the divergence of the subring zeta function of Z n and its local factors. Lastly, we apply these results to the problem of counting orders in a number field.