We prove that, analogous to the Hilbert–Kunz density function, (used for studying the Hilbert–Kunz multiplicity, the leading coefficient of the Hibert–Kunz function), there exists a $$\beta $$-density function $$g_{R, \mathbf{m}}:[0,\infty )\longrightarrow {\mathbb {R}}$$, where $$(R, \mathbf{m})$$ is the homogeneous coordinate ring associated with the toric pair (X, D), such that $$\begin{aligned} \int _0^{\infty }g_{R, \mathbf{m}}(x)\mathrm{d}x = \beta (R, \mathbf{m}), \end{aligned}$$where $$\beta (R, \mathbf{m})$$ is the second coefficient of the Hilbert–Kunz function for $$(R, \mathbf{m})$$, as constructed by Huneke–McDermott–Monsky. Moreover, we prove, (1) the function $$g_{R, \mathbf{m}}:[0, \infty )\longrightarrow {\mathbb {R}}$$ is compactly supported and is continuous except at finitely many points, (2) the function $$g_{R, \mathbf{m}}$$ 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.
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