This paper is concerned with the Hawkins random sieve which is a probabilistic analogue of the sieve of Eratosthenes. Analogues of the prime number theorem, Mertens' theorem and the Riemann hypothesis have previously been established for the Hawkins sieve. In the present paper we give a more delicate analysis using iterated logarithm results for both martingales and tail sums of martingale differences to deduce a considerably improved $\log\log$ replacement for the Riemann hypothesis result.
"A Log Log Improvement to the Riemann Hypothesis for the Hawkins Random Sieve." Ann. Probab. 6 (5) 870 - 875, October, 1978. https://doi.org/10.1214/aop/1176995433