Mathematical Proceedings of the Cambridge Philosophical Society

We improve the current upper and lower bounds for the normal order of the Erdos-Hooley Delta-function Delta(n) := sup(u is an element of R) Sigma(d vertical bar n0<log d-u <= 1) 1 (n is an element of N*), obtaining, for almost all integers n, the inequalities (log(2) n)(gamma+0(1)) < Delta(n) < (log(2) n)(log 2+0(1)) where the exponent gamma := (log 2)/log((1 - 1/log 27)/(1 - 1/log 3)) approximate to 0.33827 is conjectured to be optimal.