On critical small intervals containing primes

TL;DR

This paper investigates the probability of primes occurring within specific small intervals around primes, introducing pseudo-Ramanujan primes and establishing a lower bound of 0.5 for this probability.

Contribution

It defines a new probabilistic framework using pseudo-Ramanujan primes and analyzes prime distribution within critical small intervals.

Findings

Probability of primes in (p, 2p_{n+1}) is at least 0.5 if such probability exists.

Introduces pseudo-Ramanujan primes to study prime distribution.

Connects findings with OEIS sequence A080359.

Abstract

Let p be an odd prime, such that p_n<p/2<p_{n+1}, where p_n is the n-th prime. We study the following question: with what probability does there exist a prime in the interval (p, 2p_{n+1})? After the strong definition of the probability with help of the Ramanujan primes ([11], [12])and the introducing pseudo-Ramanujan primes, we show, that if such probability P exists, then P>=0.5. We also study a symmetrical case of the left intervals, which connected with sequence A080359 at OEIS.