Timeline for The Correlation of the Möbius Function and Dirichlet Characters
Current License: CC BY-SA 3.0
12 events
| when toggle format | what | by | license | comment | |
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| Dec 3, 2011 at 9:32 | vote | accept | Eric Naslund | ||
| Dec 3, 2011 at 9:26 | comment | added | Alan Haynes | @Greg Your second comment is relevant but not as crucial. Certainly if you have an odd character then what I said about the symmetry is true. Otherwise you can still argue the same way- the worst case scenario is to have as many −1's as possible, but by orthogonality this can be at most 1/2 of the residue classes. I'll edit this to make it (hopefully) correct. Thanks for your comments, it is a fun problem. | |
| Dec 3, 2011 at 9:23 | history | edited | Alan Haynes | CC BY-SA 3.0 |
Incorporated Greg's comments
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| Dec 3, 2011 at 9:19 | comment | added | Alan Haynes | I agree with your first comment- I thought about that before I went to sleep last night and I think you are correct. I believe what you claim, but would you mind explaining how to construct a Dirichlet character with the property you mention? | |
| Dec 3, 2011 at 9:16 | history | edited | Alan Haynes | CC BY-SA 3.0 |
added 264 characters in body
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| Dec 3, 2011 at 8:33 | vote | accept | Eric Naslund | ||
| Dec 3, 2011 at 9:30 | |||||
| Dec 3, 2011 at 8:20 | comment | added | Greg Martin | (Also I don't understand what you mean by "the values of $\chi(d)$ are symmetric about the imaginary axis". If $\chi$ is a cubic character then its values are $-\frac12\pm\frac{\sqrt{3}}2i$ and $1$, which does not have that symmetry; the same is true of any odd-order character.) | |
| Dec 3, 2011 at 8:18 | comment | added | Greg Martin | One needs to be careful here. I think this is a valid proof that $\phi_\chi(n)/n \ll_\chi \sqrt{\log\log n}$; however, the implicit constant will depend upon $\chi$. Given $n$, one can find a Dirichlet character $\chi$ such that all the small primes $p$ have $\chi(p)=-1$. The resulting construction shows that no bound better than $\phi_\chi(n)/n \ll \log\log n$ can hold uniformly over $n$ and $\chi$. | |
| Dec 3, 2011 at 7:17 | vote | accept | Eric Naslund | ||
| Dec 3, 2011 at 8:20 | |||||
| Dec 2, 2011 at 23:51 | history | edited | Alan Haynes | CC BY-SA 3.0 |
Minor touch-ups and clarification
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| Dec 2, 2011 at 23:21 | history | edited | Alan Haynes | CC BY-SA 3.0 |
added 68 characters in body
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| Dec 2, 2011 at 22:49 | history | answered | Alan Haynes | CC BY-SA 3.0 |