Density function for the second coefficient of the Hilbert-Kunz function

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We prove that, analogous to the HK density function, (used for studying the Hilbert-Kunz multiplicity, the leading coefficient of the HK function), there exists a β-density function gR, m:[0,∞) R, where (R, m) is the homogeneous coordinate ring associated to the toric pair (X, D), such that ∫₀∞gR, m(x)dx = β(R, m), where β(R, m) is the second coefficient of the Hilbert-Kunz function for (R, m), as constructed by Huneke-McDermott-Monsky. Moreover we prove, (1) the function gR, m:[0, ∞) R is compactly supported and is continuous except at finitely many points, (2) the function gR, m is multiplicative for the Segre products with the expression involving the first two coefficients of the Hilbert polynomials of the rings involved. Here we also prove and use a result (which is a refined version of a result by Henk-Linke) on the boundedness of the coefficients of rational Ehrhart quasi-polynomials of convex rational polytopes.