Let \(M\) be a finitely generated module over a local noetherian ring \((R,m)\) of characteristic \(p > 0\), and let \(I\) be an \(m\)-primary ideal. Let \(q = p^n\) and \(I^{[q]} = (i^q : i \in I)\). The Hilbert-Kunz function of \(M\) is the function \(n \mapsto l_n (M) = l_R (M/I^{[q]}M)\).
Some results of the paper: If \(R = k[[X,Y]]\), then \(l_n (M) = (\text{rk}M) p^{2n} + Cp^n + \Delta_n\), where \(C\) is an integer \(\geq 0\) and \(\Delta_n\) is an eventually periodic function of \(n\). If \(R = k[[X_1, \ldots, X_d]]\), \(d \geq 3\), then \(l_n (M) = (\text{rk} M)p^{dn} + Cp^{(d - 1)n} + O(p^{(d - 2)n})\), where \(C\) is a real number \(\geq 0\). Several open problems are stated.