Hilbert-kunz density function and hilbert-kunz multiplicity

For a pair (M, I), where M is a finitely generated graded module over a standard graded ring R of dimension d, and I is a graded ideal with ℓ(R/I) < ∞, we introduce a new invariant HKd(M, I) called the Hilbert-Kunz density function. We relate this to the Hilbert-Kunz multiplicity e _HK (M, I) by an integral formula. We prove that the Hilbert-Kunz density function satisfies a multiplicative formula for a Segre product of rings. This gives a formula for e _HK of the Segre product of rings in terms of the HKd of the rings involved. As a corollary, e _HK of the Segre product of any finite number of projective curves is a rational number.