Gaps Between Consecutive Primes and the Exponential Distribution
Fuente:
arXiv
Guardado en:
| Autor principal: | |
|---|---|
| Formato: | Preprint |
| Publicado: |
2024
|
| Materias: | |
| Acceso en línea: | |
| Etiquetas: |
Agregar Etiqueta
Sin Etiquetas, Sea el primero en etiquetar este registro!
|
| _version_ | 1866909222930743296 |
|---|---|
| author | Cohen, Joel E. |
| author_facet | Cohen, Joel E. |
| contents | Based on the primes less than $4 \times 10^{18}$, Oliveira e Silva et al. (2014) conjectured an asymptotic formula for the sum of the $k$th power of the gaps between consecutive primes less than a large number $x$. We show that the conjecture of Oliveira e Silva holds if and only if the $k$th moment of the first $n$ gaps is asymptotic to the $k$th moment of an exponential distribution with mean $\log n$, though the distribution of gaps is not exponential. Asymptotically exponential moments imply that the gaps asymptotically obey Taylor's law of fluctuation scaling: variance of the first $n$ gaps $\sim$ (mean of the first $n$ gaps)$^2$. If the distribution of the first $n$ gaps is asymptotically exponential with mean $\log n$, then the expectation of the largest of the first $n$ gaps is asymptotic to $(\log n)^2$. The largest of the first $n$ gaps is asymptotic to $(\log n)^2$ if and only if the Cramér-Shanks conjecture holds. Numerical counts of gaps and the maximal gap $G_n$ among the first $n$ gaps test these results. While most values of $G_n$ are better approximated by $(\log n)^2$ than by other models, seven values of $n$ with $G_{n} >2 e^{-γ}(\log n)^2$ suggest that $\limsup_{n \to\infty} G_n/[2e^{-γ}(\log n)^2]$ may exceed 1. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_16019 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Gaps Between Consecutive Primes and the Exponential Distribution Cohen, Joel E. Number Theory primary 11N05, secondary 11K31 Based on the primes less than $4 \times 10^{18}$, Oliveira e Silva et al. (2014) conjectured an asymptotic formula for the sum of the $k$th power of the gaps between consecutive primes less than a large number $x$. We show that the conjecture of Oliveira e Silva holds if and only if the $k$th moment of the first $n$ gaps is asymptotic to the $k$th moment of an exponential distribution with mean $\log n$, though the distribution of gaps is not exponential. Asymptotically exponential moments imply that the gaps asymptotically obey Taylor's law of fluctuation scaling: variance of the first $n$ gaps $\sim$ (mean of the first $n$ gaps)$^2$. If the distribution of the first $n$ gaps is asymptotically exponential with mean $\log n$, then the expectation of the largest of the first $n$ gaps is asymptotic to $(\log n)^2$. The largest of the first $n$ gaps is asymptotic to $(\log n)^2$ if and only if the Cramér-Shanks conjecture holds. Numerical counts of gaps and the maximal gap $G_n$ among the first $n$ gaps test these results. While most values of $G_n$ are better approximated by $(\log n)^2$ than by other models, seven values of $n$ with $G_{n} >2 e^{-γ}(\log n)^2$ suggest that $\limsup_{n \to\infty} G_n/[2e^{-γ}(\log n)^2]$ may exceed 1. |
| title | Gaps Between Consecutive Primes and the Exponential Distribution |
| topic | Number Theory primary 11N05, secondary 11K31 |
| url | https://arxiv.org/abs/2405.16019 |