Timeline for $\pi((n+1)^2)-\pi(n^2) \le \pi(n)$ for all $n \ge 370$?
Current License: CC BY-SA 4.0
5 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jul 6, 2018 at 11:08 | comment | added | juan | I don't know. Our knowledge make the assertion unlikely. | |
| Jul 6, 2018 at 10:44 | comment | added | Đào Thanh Oai | I am sorry, could you answer that the limmit is true or not true? | |
| Jul 6, 2018 at 9:54 | comment | added | juan | If you substitute each $\pi(\cdot)$ by the corresponding $\textrm{Li}(\cdot)$, the resulting function tends to $1$. But the difference $\pi(x)-\textrm{Li}(x)$ has been shown to be $\Omega_{\pm}(\frac{x^{1/2}\log\log\log x}{\log x})$. Therefore one of the error $\pi(x^2)-\textrm{Li}(x^2)$ can be of order $\frac{x}{\log x}\log\log\log x$. Therefore the sup limit of this divided by $\pi(x)$ is $+\infty$. So only if the error in $\pi((x+1)^2)$ and $\pi(x^2)$ collaborate can the limit be $1$. | |
| Jul 6, 2018 at 9:16 | comment | added | Đào Thanh Oai | Do You think $\lim_{n \to +\infty } \frac{\pi((n+1)^2)-\pi(n^2)}{\pi(n)}=1$ ?? | |
| Jul 6, 2018 at 8:57 | history | answered | juan | CC BY-SA 4.0 |