Timeline for Do Abel summation and zeta summation always coincide?
Current License: CC BY-SA 2.5
10 events
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| Apr 30, 2010 at 19:58 | comment | added | Brad Rodgers | Alright. I think I fixed everything. I did finally take a look at Hardy; I'm not sure if his proof is properly 'homotopic' to this one, but the main idea is still to take Mellin transforms. | |
| Apr 30, 2010 at 19:55 | history | edited | Brad Rodgers | CC BY-SA 2.5 |
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| Apr 29, 2010 at 19:29 | comment | added | Brad Rodgers | Actually, I had another think over this today, and I'm making a mistake in using dominated convergence. If nothing else, one (I think) can get by establishing the interchange of summation and integration for large enough $s$, and then invoke analytic continuation, but the result ought to have a real variable proof. Perhaps until I've thought about it more thoroughly, Gerald's reference to Hardy is the place to go... | |
| Apr 29, 2010 at 10:45 | vote | accept | David E Speyer | ||
| Apr 28, 2010 at 19:14 | history | edited | Brad Rodgers | CC BY-SA 2.5 |
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| Apr 28, 2010 at 19:11 | comment | added | Brad Rodgers | You're right; it's a nontrivial point. I added an argument to the post justifying it. | |
| Apr 28, 2010 at 19:06 | history | edited | Brad Rodgers | CC BY-SA 2.5 |
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| Apr 28, 2010 at 12:23 | comment | added | David E Speyer | Possibly dumb question: are you sure we can interchange summation and integration? The obvious fact is that $\sum a_n/n^s = (1/\Gamma(s)) \sum \int \cdots$, while your argument needs $\int \sum$. | |
| Apr 28, 2010 at 10:11 | history | edited | Brad Rodgers | CC BY-SA 2.5 |
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| Apr 28, 2010 at 8:41 | history | answered | Brad Rodgers | CC BY-SA 2.5 |