The number of primes in short intervals and numerical calculations for Harman's sieve

TL;DR

The paper advances the problem of finding primes in very short intervals by establishing nontrivial bounds for the count of primes in intervals of length $x^{\theta}$ with $0.52\le\theta\le 0.525$, improving upon prior barriers via Harman’s sieve and refined Buchstab-decomposition techniques. It develops detailed sieve-asymptotic formulas for complex multi-variable sums, leveraging Type I/II information and mean-value theorems for Dirichlet polynomials, and introduces new arithmetic inputs to optimize high-dimensional losses. The author then performs two large-scale decompositions to obtain explicit lower and upper bounds at $\theta=0.52$: a positive lower bound $\mathbf{LB}(0.52)\frac{x^{0.52+\varepsilon}}{\log x}$ (with $\mathbf{LB}(0.52)>0.004$) and an upper bound $\mathbf{UB}(0.52)\frac{x^{0.52+\varepsilon}}{\log x}$ (with $\mathbf{UB}(0.52)<2.874$), and discusses corresponding results for nearby $\theta$ values. These bounds validate Harman–Pintz’s approach at the $0.52$ level and yield numerous applications, including refined distribution results in arithmetic progressions, Goldbach-type statements, Carmichael-number growth, Linnik-type constants for prime powers, Sidon-set sums, and Waring–Goldbach problems in short intervals.

Abstract

The author gives nontrivial upper and lower bounds for the number of primes in the interval $[x - x^θ, x]$ for some $0.52 \leqslant θ\leqslant 0.525$, showing that the interval $[x - x^{0.52}, x]$ contains prime numbers for all sufficiently large $x$. This refines a result of Baker, Harman and Pintz (2001) and gives an affirmative answer to Harman and Pintz's argument. New arithmetic information, a delicate sieve decomposition, various techniques in Harman's sieve and accurate estimates for integrals are used to good effect.

Theorems & Definitions (29)

• Theorem 1
• Theorem 2
• Lemma 3.1
• Lemma 3.2
• Lemma 3.3
• Remark 3.4
• Remark 3.5
• Lemma 4.1
• Lemma 4.2
• proof