Overview
The research focuses on establishing effective bounds for the existence of prime numbers within short sub-linear intervals. It builds upon previous theoretical work by providing explicit conditions for these prime-containing intervals, thereby rendering certain prior results fully effective.
Research Context
The study relates to a long-standing question concerning the distribution of prime numbers. Eighty-five years prior to this work, Hoheisel demonstrated the existence of primes within the sub-linear interval $ \left[x, x+x^{1-{1\over 33000}}\right] $ for sufficiently large $x$. Heilbronn subsequently refined this by proving prime existence in the interval $ \left[x, x+x^{1-{1\over 250}}\right] $, also for sufficiently large $x$. More recently, Baker, Harman, and Pintz investigated the existence of primes in the sub-linear interval $ \left[x, x+ x^{1-{19\over 40}}\right] $ for sufficiently large $x$. While these previous works established the existence of primes in specified sub-linear intervals, the conditions for "sufficiently large $x$" were not always made explicit.
Findings
The present article describes several explicit statements regarding the existence of primes in sub-linear intervals. Specifically, the research indicates the following:
- For all $n \geq 4$ and for all $x \geq \exp(3\exp(33))$, primes exist in the interval $ \left[x, x+ x^{1-{1\over n}}\right] $.
- For all $n \geq 91$ and for all $x \geq [90^{90}]^{n/(n-90)}$, primes exist in the interval $ \left[x, x+ x^{1-{1\over n}}\right] $.
- For all $n \geq 106$ and for all $x \geq 1$, primes exist in the interval $ \left[x, x+ x^{1-{1\over n}}\right] $.
This latter observation, concerning $n \geq 106$ and $x \geq 1$, is noted to make the results originally presented by Hoheisel and Heilbronn fully explicit and effective.