A note on zero-density approaches for the difference between consecutive primes

2024/11/04 by Valeriia Starichkova, Starichkova, Valeriia · 2 citations
Mathematics · #11M26 #11N05 #Analytic Number Theory Research #FOS: Mathematics #Mathematics and Applications #Meromorphic and Entire Functions #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2411.01845

openalex publication_date 2024/11/04 · openalex created_date 2024/11/14 · openalex updated_date 2026/07/28

Abstract

In this note, we generalise two results on prime numbers in short intervals. The first result is Ingham's theorem which connects the zero-density estimates with short intervals where the prime number theorem holds, and the second result is due to Heath-Brown and Iwaniec, which derives the weighted zero-density estimates used for obtaining the lower bound for the number of primes in short intervals. The generalised versions of these results make the connections between the zero-free regions, zero-density estimates, and the primes in short intervals more transparent. As an example, the generalisation of Ingham's theorem implies that, under the Density Hypothesis, the prime number theorem holds in [x - √(x)exp(log^2/3+εx), x], which refines upon the classic interval [x - x^{1/2+ ε}, x].

Cited by

• An explicit form of Ingham's zero density estimate
• Refinements for primes in short arithmetic progressions

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