Timeline for Where does the error term of the Prime Number Theorem touch the predicted asymptotic behavior
Current License: CC BY-SA 3.0
13 events
| when toggle format | what | by | license | comment | |
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| Jul 20, 2011 at 1:28 | comment | added | paul garrett | Hm! So RH is somewhat more potentially negate-able than I'd thought! Not that I expect to do so... :) | |
| Jul 19, 2011 at 22:16 | comment | added | GH from MO | Once I verified that Schoenfeld's result implies $|\pi(x)-li(x)|<x^{1/2}\log x$ for all $x>2$. | |
| Jul 19, 2011 at 21:31 | comment | added | Stopple | @Paul, I just looked up the Schoenfeld result: On RH, $|\pi(x)-li(x)|$ is less than $x^{1/2}\log(x)/(8\pi)$ for $2,657\le x$, no mention that this is optimal. | |
| Jul 19, 2011 at 21:07 | comment | added | paul garrett | First, in response to @Stopple's recollection: if this is true everywhere, not merely asymptotically, then it is testable!?! As to the meaning of big-Oh, it means bounded by a constant multiple of... | |
| Jul 19, 2011 at 20:52 | comment | added | André Henriques | I'm confused by all your comments above. The big-O notation means that the error term is bounded by $\sqrt{x}\log(x)$, not that it's asymptotically proportional to it. See en.wikipedia.org/wiki/Big_O_notation. | |
| Jul 19, 2011 at 20:52 | answer | added | David E Speyer | timeline score: 7 | |
| Jul 19, 2011 at 20:33 | comment | added | Stopple | On the Riemann Hypothesis, Schoenfeld showed the constant $C$ can be taken as $1/(8\pi)$, and (iirc) this can not be improved on. | |
| Jul 19, 2011 at 20:21 | answer | added | GH from MO | timeline score: 12 | |
| Jul 19, 2011 at 20:04 | comment | added | user16557 | Oh ok I was under the misimpression that the constant was known. | |
| Jul 19, 2011 at 20:03 | history | undeleted | user16557 | ||
| Jul 19, 2011 at 20:03 | history | deleted | user16557 | ||
| Jul 19, 2011 at 19:58 | comment | added | Qiaochu Yuan | No, one would expect it would come near $C \sqrt{x} \log x$ for an appropriate constant. | |
| Jul 19, 2011 at 19:57 | history | asked | user16557 | CC BY-SA 3.0 |