Timeline for Reference requested for $\lim_{n \rightarrow \infty} \frac{\sum_{i=1}^{n} \bar{s}(i)}{n^2} = \frac{\pi^2}{30}$
Current License: CC BY-SA 3.0
10 events
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| Jun 13, 2011 at 16:50 | comment | added | Granger | @GH - I think it is standard etiquette (and saves space) to determine whether or not a result is already in print, before contributing a proof. Therefore whether a result has been considered before is important. I think you and Junkie misunderstood what I was asking. As it is, writing `this is a standard exercise' would have been a sufficient answer, so thank you. | |
| Jun 11, 2011 at 4:20 | comment | added | GH from MO | This is a standard exercise in analytic number theory, regardless if it had been considered before. You got a solution, use it and enjoy it. It is possible that there is a more elementary proof, using convolutions and the hyperbola method. | |
| Jun 9, 2011 at 9:35 | comment | added | Granger | Thanks guys, but I was only asking if to anyones knowledge this had been considered before (hence reference request). It's sufficiently natural that I thought it must be classical? | |
| Jun 8, 2011 at 23:28 | comment | added | Junkie | You are right, I was misreading the intent of the question as $\sum_n \bar s(n)/n^2$, and was getting a divergent sum. | |
| Jun 8, 2011 at 23:27 | history | edited | Junkie | CC BY-SA 3.0 |
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| Jun 8, 2011 at 23:14 | comment | added | GH from MO | Junkie, at the end of your solution you still say "Partial summation then gives the indicated result". There is no partial summation here, your asymptotic formula is the limit in question: the left hand side divided by $X^2$ tends to $\frac{\pi^2}{30}$, period. Also convergence is no issue if you couple the points $s$ in conjugate pairs. | |
| Jun 8, 2011 at 22:59 | history | edited | Junkie | CC BY-SA 3.0 |
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| Jun 8, 2011 at 22:56 | comment | added | Junkie | Ah yes, you are right, and really there should be more fretting about convergence. | |
| Jun 8, 2011 at 22:41 | comment | added | GH from MO | Junkie, by Perron's formula we have $$\sum_{n\le X} \bar s(n)=\int_{(3)}\frac{\zeta(2s)\zeta(s-1)}{\zeta(2s-2)}\frac{X^s}{s}\,ds\sim \frac{\pi^2}{30}X^2,$$ which is the limit conjectured by the OP. So there is no need to do any partial summation: the extra factor $2$ comes from the $s$ in the denominator of the integrand. | |
| Jun 8, 2011 at 21:51 | history | answered | Junkie | CC BY-SA 3.0 |