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 [n²,(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.