Depth of associated graded rings via Hilbert coefficients of ideals

Given a local Cohen-Macaulay ring $(R,\mathfrak{m})$, we study the interplay between the integral closedness – or even the normality – of an $\mathfrak{m}$-primary $R$-ideal $I$ and conditions on the Hilbert coefficients of $I$. We relate these properties to the depth of the associated graded ring of $I$.