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

Density function for the second coefficient of the Hilbert–Kunz function on projective toric varieties