Primes in Short Intervals: Heuristics and Calculations

Abstract

We formulate, using heuristic reasoning, conjectures for the range of the number of primes in intervals of length y around x, where $y\ll {(\hspace{0.17em}\text{log}\hspace{0.17em}x)}^2$. In particular, we conjecture that the maximum grows surprisingly slowly as y ranges from $\text{log}\hspace{0.17em}x$ to ${(\hspace{0.17em}\text{log}\hspace{0.17em}x)}^2$. We will exhibit the available data, showing that it somewhat supports our conjectures, though not so well that there may not be room for some modifications.

Dedicated to the memory of Lord Cherwell.*

Keywords:

• primes
• short intervals
• heuristics
• probability distributions

Declaration of Interest

No potential conflict of interest was reported by the author(s).

Acknowledgments

Thanks are due to James Maynard for some helpful remarks on both the content and the exposition, to Kevin Ford and Drew Sutherland for making various data available, as well as to the three anonymous referees for their helpful comments. The first author was supported by a Canadian Research Chair and the second author was supported by an NSERC fellowship for the duration of this project.

Additional information

Funding

Thanks are due to James Maynard for some helpful remarks on both the content and the exposition, to Kevin Ford and Drew Sutherland for making various data available, as well as to the three anonymous referees for their helpful comments. The first author was supported by a Canadian Research Chair and the second author was supported by an NSERC fellowship for the duration of this project.