Timeline for Density of numbers not divisible by a large prime power
Current License: CC BY-SA 2.5
9 events
| when toggle format | what | by | license | comment | |
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| Oct 6, 2018 at 11:05 | vote | accept | Woett | ||
| Apr 13, 2017 at 12:57 | history | edited | CommunityBot |
replaced http://mathoverflow.net/ with https://mathoverflow.net/
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| Mar 31, 2011 at 20:30 | comment | added | GH from MO | $z=\max(17,b)$ is fine, I was lazy typing and had to fit my message in a bounded box :-) Also I made no attempt to optimize any of my estimates. Perhaps I should add that the original problem naturally reduces to square-free numbers as the density of numbers with a square divisor exceeding $u^2$ is less than $1/u$ (think of $u$ as a large constant). Actually, this is how I was led to my solution. | |
| Mar 31, 2011 at 19:53 | comment | added | Woett | Thanks a lot! Not that it matters much, but isn't it for large $b$ better (in the sense that, when $b$ is actually known, we get a better lower bound on the number of good integers) to choose $z$ to be $\max{17, b}$ instead of $> 5b$? | |
| Mar 31, 2011 at 19:20 | comment | added | GH from MO | Here is how to deduce the statement for any residue class $a \pmod{b}$ using my construction. Choose $z > 5b$, then the construction yields a set of good integers $x < n \leq 2x$ coprime with $b$, which is of positive lower density. This set intersects at least one reduced residue class $c \pmod{b}$ with positive lower density. Multiply this intersection with the residue $1 \leq a\bar c \leq b$, where $\bar c$ stands for multiplicative inverse mod $b$. For $x > b^2$ the result is a set of good integers $x < n \leq 2bx$ in the residue class $a \pmod{b}$, which is of positive lower density. | |
| Mar 31, 2011 at 17:51 | comment | added | GH from MO | Woett, I believe my proof can be easily adapted to residue classes, but I am lazy to check all the details. | |
| Mar 31, 2011 at 17:37 | history | edited | Woett | CC BY-SA 2.5 |
added 202 characters in body
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| Mar 31, 2011 at 15:45 | answer | added | GH from MO | timeline score: 4 | |
| Mar 31, 2011 at 13:40 | history | asked | Woett | CC BY-SA 2.5 |