The expected number of elements to generate a finite group with d-generated sylow subgroups

Given a finite group G; let e(G) be the expected number of elements of G which have to be drawn at random, with replacement, before a set of generators is found. If all of the Sylow subgroups of G can be generated by d elements, then e(G) ≤ d + κ, where κ is an absolute constant that is explicitly described in terms of the Riemann zeta function and is the best possible in this context. Approximately, κ equals 2.752394. If G is a permutation group of degree n; then either G = Sym(3) and e(G) = 2:9 or e(G) ≤ {n=2}+κ with κ ∼1:606695: These results improve weaker bounds recently obtained by Lucchini.