Timeline for When is $n/\ln(n)$ close to an integer?
Current License: CC BY-SA 2.5
8 events
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| Jun 14, 2010 at 23:04 | comment | added | Wadim Zudilin | Yes, Will, but your formulation makes the things highly depend on the arithmetic of Lambert's $W$. And the latter arithmetic is not done yet... | |
| Jun 14, 2010 at 15:46 | comment | added | Will Jagy | I ought to have said, with $ n < x_0 < n+1,$ that we demand $0 < f'(n) < \epsilon,$ as $ f''(x) < 0 $ when $ x > e^2.$ The numerology question stands, when do we get especially small $ k - f(n) $ or $ f(n+1) - k.$ With my choice of symbols, we could simply ask when either $ x_0 - n$ or $n+1 - x_0$ is small. This formulation takes the size of $f'$ out of the discussion. | |
| Jun 14, 2010 at 7:41 | comment | added | Wadim Zudilin | Excellent! I am happy to see how the numerological problem is demustified. Thank you very much, Kevin! I was far from the internet, but your argument is indeed very clear. Sorry for asking you more details. | |
| Jun 14, 2010 at 7:38 | vote | accept | Wadim Zudilin | ||
| Jun 14, 2010 at 4:44 | comment | added | Will Jagy | In fact, taking that $x_0$ such that $ f(x_0) = k \in Z,$ with $ 0 < f'(x_0) < \epsilon,$ and writing $ n < x_0 < n+1,$ we get $ k < f(n+1) < k + \epsilon $ and then $ k - \epsilon < f(n) < k.$ | |
| Jun 14, 2010 at 4:33 | comment | added | Kevin Ventullo | Let $[ ]$ denote the ceiling function. Fix $\epsilon>0$. Choose a real number $x$ so that $f(x)\in \mathbb{Z}$ and choose it large enough so that $\frac{1}{\log x} - \frac{1}{(\log x)^2}<\epsilon$. Then one can show $||f([x])||<\epsilon$ by combining the Mean Value Theorem with the fact that $|x-[x]|<1$. | |
| Jun 14, 2010 at 4:18 | comment | added | Wadim Zudilin | So, if we can choose a sequence of $x$'s whose distance to $\mathbb Z$ decreases, then we have the desired result. (Maybe, you wish to use in some way the fact that $f$ is contraction, but then it's still non-obvious to me, so please expand your argument or let me to think more.) How to guarantee that $x$ is close to an integer? This would refer to arithmetic properties of Lambert-like ($W$) function which are not even conjecturally studied... | |
| Jun 14, 2010 at 3:51 | history | answered | Kevin Ventullo | CC BY-SA 2.5 |