Ratliff-Rush filtration, Hilbert coefficients and reduction number of integrally closed ideals

Let (R,m) be a Cohen-Macaulay local ring of dimension d >= 3 and I an integrally closed m-primary ideal. We establish bounds for the third Hilbert coefficient e3(I) in terms of the lower Hilbert coefficients e(i)(I),0 <= i <= 2 and the reduction number of I. When d=3, the boundary cases of these bounds characterize certain properties of the Ratliff-Rush filtration of I. These properties, though weaker than depth G(I)>= 1, guarantee that Rossi's bound for reduction number r(j)(I) holds in dimension three. In that context, we prove that if depth G(I) >= d-3, then r(j )(I) <= l1(I)-e(0)(I)+& ell;(R/I)+1+e(2)(I)(e(2)(I)-e(1)(I)+e(0)(I)-& ell;(R/I))-e(3)(I). We also discuss the signature of the fourth Hilbert coefficient e(4)(I).