Note On Prime Gaps And Very Short Intervals

TL;DR

Under the assumption of the Riemann hypothesis, the paper presents an elementary argument suggesting that prime gaps are bounded by a function involving logarithms, and very short intervals contain primes for large x.

Contribution

It introduces a new elementary approach to bound prime gaps and short intervals containing primes under the Riemann hypothesis.

Findings

Prime gaps are bounded by c1((logx)^2)/loglogx.

Very short intervals of length greater than c2((logx)^2)/loglogx contain primes.

The results depend on the Riemann hypothesis.

Abstract

Assuming the Riemann hypothesis, this article discusses a new elementary argument that seems to prove that the maximal prime gap of a finite sequence of primes p_1, p_2, ..., p_n <= x, satisfies max {p_(n+1) - p_n : p_n <= x} <= c1((logx)^2)/loglogx, c1 > 0 constant. Equivalently, it shows that the very short intervals (x, x + y] contain prime numbers for all y > c2((logx)^2)/loglogx, c2 > 0 constant, and sufficiently large x > 0.