Timeline for What results would follow from or imply "randomness" of the primes?
Current License: CC BY-SA 3.0
8 events
| when toggle format | what | by | license | comment | |
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| May 22, 2011 at 3:21 | comment | added | Peter Humphries | @Qiaochu: What I mean is that the heurestic given by Ng (see near the end of cs.uleth.ca/~nathanng/RESEARCH/mobius2b.pdf) is justified by theoretical evidence in line with other, more accepted conjectures (Riemann hypothesis, linear independence hypothesis, and discrete moments of L-functions), in comparison to other conjectured growth rates of $\sum_{n < x}{\mu(n)}$, which, as Ng discusses, can vary greatly. | |
| May 21, 2011 at 7:40 | comment | added | Qiaochu Yuan | What does it mean for a conjecture to be better than another conjecture? | |
| May 21, 2011 at 4:45 | comment | added | Douglas Zare | Since it's community wiki, I corrected the statements. | |
| May 21, 2011 at 4:43 | history | edited | Douglas Zare | CC BY-SA 3.0 |
changed estimates in first paragraph; deleted 107 characters in body
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| May 21, 2011 at 4:32 | comment | added | Peter Humphries | What's interesting here is that the best conjecture so far on the size of $\sum_{n < x}{\mu(n)}$ is by Ng, which is that these sums are only greater than $C \sqrt{x} (\log \log \log x)^{5/4}$ infinitely often, but not any function that grows faster. So there's a bit of a disparity between the probabilistic model and the real thing. | |
| May 20, 2011 at 21:35 | comment | added | Felipe Voloch | @Mark: You are probably right, thanks. | |
| May 20, 2011 at 21:25 | comment | added | Mark Lewko | I think you need a bound of $O_{\epsilon}(x^{1/2 + \epsilon})$ to be equivalent to RH. (In the probabilistic model, the law of the iterated logarithm suggests the partial sums will be greater than $O((x\ln \ln(x) )^{1/2})$ infinitely often.) | |
| May 20, 2011 at 21:08 | history | answered | Felipe Voloch | CC BY-SA 3.0 |