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Primes in almost all short intervals II

17 October 2025, Version 1
Working Paper
This content is an early or alternative research output and has not been peer-reviewed by Cambridge University Press at the time of posting.

Abstract

By combining the sieve machinery used in the author's previous paper on this topic and some new arithmetic information in Harman's monograph, the author proves that the interval $[n-n^{\frac{1}{22}+\varepsilon}, n]$ contains prime numbers for almost all $n$, improving the previous exponent $\frac{1}{21.5}$ by the author. The use of the variable role-reversal plays a crucial role in the proof.

Keywords

Prime
Sieve methods
Dirichlet polynomial

Supplementary materials

Title
Description
Actions
Title
Mathematica code for numerical calculations (1/2)
Description
This is the Mathematica code for the numerical calculations in the preprint.
Actions
Title
Mathematica code for numerical calculations (2/2)
Description
This is the Mathematica code for the numerical calculations in the preprint.
Actions
Title
Mathematica package SieveFunctions
Description
This is Galway's Mathematica package that gives values of functions FF(s) and ff(s) in our code.
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Comment number 6, Dekai Wang: Oct 23, 2025, 07:51

打不了球了GnGcFGbefZIOJYA@RunboLi

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Version History

Oct 17, 2025 Version 1

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The content is available under CC BY NC SA 4.0 [opens in a new tab]

DOI

10.33774/coe-2025-jrnjl
D O I: 10.33774/coe-2025-jrnjl [opens in a new tab]

Author’s competing interest statement

The author(s) have declared they have no conflict of interest with regard to this content

Ethics

The author(s) have declared ethics committee/IRB approval is not relevant to this content

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