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Dedicated to the memory of D. H. Lehmer) for some constant γ > 0. In part I we showed one can take γ = .643 using an approach started by Crandall  and extended by Sander . In this paper we derive bounds (1.3) using systems of difference inequalities found by Krasikov . For each k ≥ 2 there is a system ( k of such inequalities; Krasikov  used ( 2 todoi:10.2307/2153346 fatcat:piebn4g3u5fwjimdrgmtbvwaji