Prime numbers between squares

A well known conjecture about the distribution of primes asserts that between two consecutive squares there is always at least one prime number. The proof of this conjecture is out of reach at present, even under the assumption of the Riemann Hypothesis. The aim of this paper is to provide a conditional proof of the conjecture assuming a hypothesis about the behavior of Selberg's integral in short interval

A note on primes in short intervals

This paper is concerned with the number of primes in short intervals. We prove that for every $\theta>1/2$ the intervals $[x, x+x^{\theta}]$ contain the expected number of primes for $x\rightarrow \infty$, with the assumption of an heuristic hypothesis weaker than the Lindel\"{o}f hypothesi