Large Prime Gaps

I. INTRODUCTION

Let denote the nth prime, and let

denote the maximum gap between consecutive primes less than X. It is clear from the prime number theorem that

as the average gap between the prime numbers which are is . In 1931, Westzynthius proved that infinitely often, the gap between consecutive prime numbers can be an arbitrarily large multiple of the average gap, that is, as , improving upon prior result of Backlund and Brauer-Zeitz. Moreover, the strongest unconditional lower bound on is due to Ford, Green, Konyagin, Maynard, and Tao, who have shown that

for sufficiently large X, with the k-fold iterated natural logarithm of X, whereas the strongest unconditional upper bound is

a result due to Baker, Harman, and Pintz. Assuming the Riemann Hypothesis, Cramér showed that

My main theorem is the following further quantitative improvement.

Theorem 1: (Large prime gaps). For any sufficiently large X and any sufficiently small , one has

For any sufficiently large X and any sufficiently small , we have

(1) is correct when with

and when sufficiently large

Indeed, consider , consider the following limit

We try with , (1) is correct when

Theorem 2: (Large prime gaps). For any sufficiently large X, one has

For any sufficiently large , we have

and when .

Indeed, consider , consider the following limit

ACKNOWLEDGEMENT

I thank VNU University of Science for accompanying me.