Effective short intervals containing primes

TL;DR

This paper makes explicit and effective certain classical results on the existence of primes within short intervals, improving bounds and extending the range of known prime-containing intervals for large x.

Contribution

It provides explicit bounds and conditions under which primes are guaranteed in short intervals, making previous asymptotic results fully effective and extendable.

Findings

Primes exist in intervals [x, x + x^{1-1/n}] for n ≥ 4 and x ≥ exp(exp(33)).

Primes exist in intervals [x, x + x^{1-1/n}] for n ≥ 91 and x ≥ [90^{90}]^{n/(n-90)}.

Primes exist in intervals [x, x + x^{1-1/n}] for n ≥ 106 for all x ≥ 1.

Abstract

95 years ago Hoheisel proved the existence of primes in the sub-linear interval \[ \left[x, x+x^{1-{1\over 33000}}\right] \qquad \hbox{for $x$ sufficiently large}. \] This was improved by Heilbronn, proving existence of primes in the interval \[ \left[x, x+x^{1-{1\over 250}}\right] \qquad \hbox{for $x$ sufficiently large}. \] More recently Baker, Harman, Pintz proved existence of primes in the interval \[ \left[x, x+ x^{1-{19\over 40}}\right] \qquad \hbox{for $x$ sufficiently large}. \] In the present article I will, to the extent possible, make some of these statements effective. Specifically, among other things, I shall show that \[ \forall n \geq 4, \qquad\forall x \geq \exp(\exp(33)), \qquad \hbox{there are primes in the interval} \left[x, x+ x^{1-{1\over n}}\right]; \] \[ \forall n \geq 91, \qquad\forall x \geq [90^{90}]^{n/(n-90)} , \qquad \hbox{there are primes in the interval} \left[x, x+ x^{1-{1\over n}}\right]. \] Furthermore \[ \forall n \geq 106, \qquad\forall x \geq 1, \qquad \hbox{there are primes in the interval} \left[x, x+ x^{1-{1\over n}}\right]. \] In particular this last observation makes both the Hoheisel and Heilbronn results fully explicit and effective. This (relatively) specific observation can be extended and generalized in various manners.