Primes between consecutive squares and the Lindelof hypothesis

At present noone knows an unconditional proof that between two consecutive squares there is always a prime number. In a previous paper the author proved that, under the assumption of the Lindelof hypothesis, each of the intervals [n (2), (n+1)(2)] aS, [1,N], with at most O(N (E >)) exceptions, contains the expected number of primes, for every constant E > > 0. In this paper we improve the result by weakening the hypothesis in two different ways.