An explicit lower bound for large gaps between some consecutive primes

2024/04/10 by Keiju Sono, Sono, Keiju
Mathematics · #11N36 #Analytic Number Theory Research #FOS: Mathematics #Finite Group Theory Research #Limits and Structures in Graph Theory #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2404.06951

openalex publication_date 2024/04/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let p_n denote the nth prime and for any fixed positive integer k and X≥ 2, put G_k(X):=max _p _n+k≤ X min \ p_n+1-p_n, … , p_n+k-p_n+k-1 \. Ford, Maynard and Tao proved that there exists an effective absolute constant c_LG>0 such that G_k(X)≥ \fracc_LGk²(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 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≈ 2.0× 10^-17.

Related

• On the gaps between consecutive primes
• Large Gaps between Primes in Arithmetic Progressions
• Note On The Maximal Primes Gaps
• Large gaps between consecutive prime numbers containing perfect k-th\n powers of prime numbers
• On the ratio of consecutive gaps between primes