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For a given algebraically closed field $k$ of characteristic $p>0$ weconsider the set ${\mathcal C}_k$, of graded isomorphism classes of {\emstandard graded pairs} $(R, I)$, where $R$ is a standard graded ring over thefield and $I$ is a graded ideal of finite colength.  Here we give a ring homomorphism  $\Pi:\Z[{\mathcal C}_k] \longrightarrow H(\C)[X]$, where $H(\C)$ denotes thering of entire functions.  The related entire function and the homomorphism $\Pi$ keep track of the two  positive characteristic invariants, $e_{HK}(R, I)$ and $c^I({\bf m})$ of thering: (1) composing the map $\Pi$ with the evaluation map at $z=0$ gives a ringhomomorphism  $\Pi_e:\Z[{\mathcal C}_k] \longrightarrow \R[X]$ which sends $$(R,I) \toe_{HK}(R^0, IR^0)+ e_{HK}(R^1, IR^1)X+\cdots + e_{HK}(R^d, IR^d)X^d,$$ where$R^i$ is the union of $i$ dimensional components of $R$ and $e_{HK}(R^i, IR^i)$is the HK multiplicity of the pair $(R^i, IR^i)$, and in particular the topcoefficient is $e_{HK}(R, I)$.  (2) If, in addition, $R$ is a two dimensional ring or $\mbox {Proj~R}$ isstrongly $F$-regular, then the Fourier transform ${\widehat f}_{R, I}$  belongs to the Paley-Wiener class of the real number, namely  the $F$-threshold $c^I_{\bf m}(R)$ of the maximal ideal ${\bf m}$.

HK multiplicity, $F$-threshold and the Paley-Wiener theorem