论文标题
$π（x）$和$π（（（x+1）^{2}）的边界的估计值 - π（x^{2}）$
Estimates of the bounds of $π(x)$ and $π((x+1)^{2})-π(x^{2})$
论文作者
论文摘要
我们使用分析数理论中的原理在质数计数函数上显示以下界限$π（x）$，给出一个估计：$$ 2 \ log 2 \ geq \ limsup_ {x \ rightarrow \ rightarrow \ infty} \ frac {π（π（x）} \ frac {π（x）} {x / \ log x} \ geq \ geq \ log 2 $$所有$ x $都足够大。我们还猜想了$π（（x+1）^{2}） - π（x^{2}）$的边界，与Legendre在上述间隔中对素数的猜想相关，以便：$$ \ feft： \ lfloor \ frac {1} {2} \ left（\ frac {\ left（x+1 \右） x \ right）^{2}} {\ log \ left（\ log x \ right）} \ right \ rfloor \leqπ（（x+1）^{2}）^{2}） - π（x^{2}） \ lfloor \ frac {1} {2} \ left（\ frac {\ left（x+1 \右） \ rfloor $$
We show the following bounds on the prime counting function $π(x)$ using principles from analytic number theory, giving an estimate: $$2 \log 2 \geq \limsup_{x \rightarrow \infty} \frac{π(x)}{x / \log x} \geq \liminf_{x \rightarrow \infty} \frac{π(x)}{x / \log x} \geq \log 2$$ for all $x$ sufficiently large. We also conjecture about the bounding of $π((x+1)^{2}) - π(x^{2})$, as is relevant to Legendre's conjecture about the number of primes in the aforementioned interval such that: $$ \left \lfloor\frac{1}{2}\left(\frac{\left(x+1\right)^{2}}{\log\left(x+1\right)}-\frac{x^{2}}{\log x}\right)-\frac{\left(\log x\right)^{2}}{\log\left(\log x\right)}\right \rfloor \leq π((x+1)^{2}) - π(x^{2}) \leq $$ $$ \left \lfloor\frac{1}{2}\left(\frac{\left(x+1\right)^{2}}{\log\left(x+1\right)}-\frac{x^{2}}{\log x}\right) + \log^{2}x\log\log x \right \rfloor$$
