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In this paper we find the tight closure of powers of parameter ideals of certain diagonal hypersurface rings. In many cases the associated graded ring with respect to tight closure filtration turns out to be Cohen-Macaulay. This helps us find the tight Hilbert polynomial in these diagonal hypersurfaces. We determine the tight Hilbert polynomial in the following cases: (1) F -pure diagonal hypersurfaces where number of variables is equal to the degree of defining equation, (2) diagonal hypersurface rings where characteristic of the ring is one less than the degree of defining equation and (3) quartic diagonal hypersurface in four variables.

Let (R, m) be a Noetherian local ring of prime characteristic p and Q be an m-primary parameter ideal. We give criteria for F-rationality of R using the tight Hilbert function. We obtain a lower bound for the tight Hilbert function of Q for equidimensional excellent local rings that generalises a result of Goto and Nakamura.We show that if dim R = 2, the Hochster-Huneke graph of R is connected and this lower bound is achieved then R is F-rational. Craig Huneke asked if the F -rationality of unmixed local rings may be characterized by the vanishing of e * 1 (Q). We construct examples to show that without additional conditions, this is not possible. Let R be an excellent, reduced, equidimensional Noetherian local ring and Q be generated by parameter test elements. We find formulas for e * 1 (Q), e * 2 (Q), . . . , e * d (Q) in terms of Hilbert coefficients of Q, lengths of local cohomology modules of R, and the length of the tight closure of the zero submodule of H d m (R). Using these we prove: R is F-rational ⇐⇒ e * 1 (Q) = e 1 (Q) ⇐⇒ depth R ≥ 2 and e * 1 (Q) = 0.

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