  Let $(R,\mathfrak m)$ be an analytically unramified local ring of positive
prime characteristic $p.$ For an ideal $I$, let $I^*$ denote its tight closure.
We introduce the tight Hilbert function $H^*_I(n)=\ell(R/(I^n)^*)$ and the
corresponding tight Hilbert polynomial $P_I^*(n)$ where $I$ is an $\mathfrak
m$-primary ideal. It is proved that $F$-rationality can be detected by the
vanishing of the first coefficient of $P_I^*(n).$ We find the tight Hilbert
polynomial of certain parameter ideals in hypersurface rings and
Stanley-Reisner rings of simplicial complexes.
