$\beta$-density function on the class group of projective toric varieties

We prove the existence of a compactly supported, continuous (except at finitely many points) function $g_{I, {\bf m}}: [0, \infty)\longrightarrow $\mathbb{R}$$forallmonomialprimeideals$I$of$R$ofheightonewhere$(R, {\bf m})$ is the homogeneous coordinate ring associated to a projectively normal toric pair $(X,D)$, such that $${\int }_0^{\mathrm{\infty }}{g}_{I,\mathbf{m}}(\lambda )d\lambda =\beta (I,\mathbf{m}),$$ where $\beta (I,\mathbf{m})$ is the second coefficient of the Hilbert-Kunz function of $I$ with respect to the maximal ideal $\mathbf{m}$, as proved by Huneke-McDermott-Monsky \cite{HMM2004}. Using the above result, for standard graded normal affine monoid rings we give a complete description of the class map $\tau_{{\bf m}}:$\text{Cl}$®\longrightarrow $\mathbb{R}$$ introduced in \cite{HMM2004} to prove the existence of the second coefficient of the Hilbert-Kunz function. Moreover, we show the function $g_{I,\mathbf{m}}$ is multiplicative on Segre products with the expression involving the first two coefficients of the Hilbert polynomial of the rings and the ideals. \keywords{coefficients of Hilbert-Kunz function\and projective toric variety\and Hilbert-Kunz density function\and $\beta$-density function\and monomial prime ideal of height one.}