Explicit lower bound for large gaps between some consecutive primes

Keiju Sono

Published 2024-04-10Version 1

Let $p_{n}$ denote the $n$th prime and for any fixed positive integer $k$ and $X\geq 2$, put \[ G_{k}(X):=\max _{p _{n+k}\leq X} \min \{ p_{n+1}-p_{n}, \ldots , p_{n+k}-p_{n+k-1} \}. \] Ford, Maynard and Tao proved that there exists an effective abosolute constant $c_{LG}>0$ such that \[ G_{k}(X)\geq \frac{c_{LG}}{k^{2}}\frac{\log X \log \log X \log \log \log \log X}{\log \log \log X} \] holds for any sufficiently large $X$. The main purpose of this paper is to determine the constant $c_{LG}$ above. We see that $c_{LG}$ is determined by several factors related to analytic number theory, for example, the level of distribution of primes, the ratio of integrals of functions in the multidimensional sieve of Maynard, the distribution of primes in arithmetic progressions to large moduli, and the coefficient of upper bound sieve of Selberg. We prove that the above inequality is valid at least for $c_{LG}\approx 3.0\times 10^{-17}$.