Timeline for Explicit bounds on number of primes of given size
Current License: CC BY-SA 4.0
8 events
| when toggle format | what | by | license | comment | |
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| Nov 22, 2022 at 15:19 | vote | accept | Bruno | ||
| Nov 21, 2022 at 23:28 | comment | added | GH from MO | For $b\geq 31$, the value of $\alpha $ lies between $1.4$ and $1/\ln 2$. See my response below. | |
| Nov 21, 2022 at 23:15 | history | edited | GH from MO |
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| Nov 21, 2022 at 23:14 | answer | added | GH from MO | timeline score: 8 | |
| Nov 21, 2022 at 12:46 | history | edited | Bruno | CC BY-SA 4.0 |
added 56 characters in body
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| Nov 21, 2022 at 12:44 | comment | added | Bruno | I have to say I am particularly embarrassed by my second question, added afterwards while it is clearly false... But actually, once you write it, your remark on the limit is obvious too! | |
| Nov 21, 2022 at 11:00 | comment | added | Wojowu | PNT easily implies that $\frac{\pi(2x)-\pi(x)}{x/\log x}$ tends to $1$ as $1\to\infty$. For $x=2^b$ you will get that your $\alpha$ tends to $\frac{1}{\ln 2}\approx 1.442$. Any version of PNT with explicit error term should let you give explicit bounds on how fast it converges. This should easily answer first and third question. For the second one, note that your table itself contains two counterexamples, consider values at $b=8.9$ or $b=11,12$. The asymptotic behavior is less clear, and I'm not sure we should expect it's eventually monotone. | |
| Nov 21, 2022 at 10:14 | history | asked | Bruno | CC BY-SA 4.0 |