**Danilo Bazzanella**

*Conditional results about primes between consecutive powers*

**Pages:** 61-69

**Received:** 14 March 2012

**Accepted:** 29 June 2012

**Mathematics Subject Classification (2010):** 11NO5.

**Keywords:** Prime numbers between powers, primes in short intervals.

**Abstract:**
A well known conjecture about the distribution of primes asserts that all intervals of type
[n^{2},(n+1)^{2}] contain at least one prime. The proof of this conjecture is
quite out of reach at present, even under the assumption of the Riemann Hypothesis.
In a previous paper the author, assuming the
Lindelöf hypothesis, proved that each of the interval [n^{α}, (n+1)^{α}]
contains the expected number of primes for α > 2 and n → ∞.
In this paper we prove the same
result assuming in turn two different heuristic hypotheses. It must be stressed that both the hypotheses
are implied by the Lindelöf hypothesis.

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