Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2011
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/1110.2952 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866909560442191872 |
|---|---|
| author | Tan, Shan-Guang |
| author_facet | Tan, Shan-Guang |
| contents | Two topics of the number theory are discussed in this paper.
First, we prove that given each natural number $x\geq10^{3}$, we have \[ |{\rm Li}(x)-π(x)|\leq c\sqrt{x}\log x\texttt{ and } π(x)={\rm Li}(x)+O(\sqrt{x}\log x) \] where $c$ is a constant greater than $1$ and less than $e$.
Second, with a much more accurate estimation of prime numbers, the error range of which is less than $x^{1/2-0.0327283}$ for $x\geq10^{41}$, we prove a theorem of the number of primes in short intervals: Given a positive real number $β$ that determines a real number $x_β$ by $e(\log x_β)^{3}/x_β^{0.0327283}=β$, let $Φ(x):=βx^{1/2}$ for $x\geq x_β$ where $Φ(x):=x^{1/2}$ when let $β=1$. Then there are \[ \frac{π(x+Φ(x))-π(x)}{Φ(x)/\log x}=1+O(\frac{1}{\log x}) \] and \[ \lim_{x \to \infty}\frac{π(x+Φ(x))-π(x)}{Φ(x)/\log x}=1. \] |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1110_2952 |
| institution | arXiv |
| publishDate | 2011 |
| record_format | arxiv |
| spellingShingle | On $|{\rm Li}(x)-π(x)|$ and primes in short intervals Tan, Shan-Guang General Mathematics 11A41, 11M26 Two topics of the number theory are discussed in this paper. First, we prove that given each natural number $x\geq10^{3}$, we have \[ |{\rm Li}(x)-π(x)|\leq c\sqrt{x}\log x\texttt{ and } π(x)={\rm Li}(x)+O(\sqrt{x}\log x) \] where $c$ is a constant greater than $1$ and less than $e$. Second, with a much more accurate estimation of prime numbers, the error range of which is less than $x^{1/2-0.0327283}$ for $x\geq10^{41}$, we prove a theorem of the number of primes in short intervals: Given a positive real number $β$ that determines a real number $x_β$ by $e(\log x_β)^{3}/x_β^{0.0327283}=β$, let $Φ(x):=βx^{1/2}$ for $x\geq x_β$ where $Φ(x):=x^{1/2}$ when let $β=1$. Then there are \[ \frac{π(x+Φ(x))-π(x)}{Φ(x)/\log x}=1+O(\frac{1}{\log x}) \] and \[ \lim_{x \to \infty}\frac{π(x+Φ(x))-π(x)}{Φ(x)/\log x}=1. \] |
| title | On $|{\rm Li}(x)-π(x)|$ and primes in short intervals |
| topic | General Mathematics 11A41, 11M26 |
| url | https://arxiv.org/abs/1110.2952 |