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

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 $\beta$-density function $g_{R, {\bf m}}:[0,\infty)\longrightarrow {\mathbb R}$, where $(R, {\bf m})$ is the homogeneous coordinate ring associated to the toric pair $(X, D)$, such that $$\int_0^{\infty}g_{R, {\bf m}}(x)dx = \beta(R, {\bf m}),$$ where $\beta(R, {\bf m})$ is the second coefficient of the Hilbert-Kunz function for $(R, {\bf m})$, as constructed by Huneke-McDermott-Monsky. Moreover we prove, (1) the function $g_{R, {\bf m}}:[0, \infty)\longrightarrow {\mathbb R}$ is compactly supported and is continuous except at finitely many points, (2) the function $g_{R, {\bf m}}$ 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.