LARGE DEVIATIONS IN THE RANDOM SIEVE

The proportion rho(k) of gaps with length k between square-free number s is shown to satisfy log rho(k)=-(1+o(1))(6/pi(2))k log k as k-->infi nity. Such asymptotics are consistent with Erdos's challenge to prove that the gap following the square-free number t is smaller than c log t/log log t, for all t and some constant c satisfying c>n(2)/12. The r esults of this paper are achieved by studying the probabilities of lar ge deviations in a certain 'random sieve', for which the proportions r ho(k) have representations as probabilities. The asymptotic form of rh o(k) may be obtained in situations of greater generality, when the squ ared primes are replaced by an arbitrary sequence (s(r)) of relatively prime integers satisfying Sigma(r)1/s(r)<infinity, subject to two fur ther conditions of regularity on this sequence.