Timeline for On the equation $\zeta(s) = F(s)+F(s+1)$
Current License: CC BY-SA 4.0
16 events
| when toggle format | what | by | license | comment | |
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| Aug 29, 2022 at 15:56 | comment | added | Dan Romik | @DavidFarmer time will tell if these ideas are useful or not, I don’t have a strong a priori opinion about this. Thanks for your interesting take in any case. | |
| Aug 29, 2022 at 15:54 | comment | added | Dan Romik | @SylvainJULIEN if you replace $T$ by $\frac{1}{2}T$ (which is a more correct scaling from the point of view of transfer operators, see my edit) then it has the constant functions as its only fixed points, and the Bernoulli polynomials as its eigenfunctions. | |
| Aug 29, 2022 at 15:51 | history | edited | Dan Romik | CC BY-SA 4.0 |
Fixed two typos; added 28 characters in body
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| Aug 29, 2022 at 13:14 | comment | added | David Farmer | @DanRomik Here is another example of a functional relation which initially seems like it might be useful, but is not: every function is the sum of an even function and an odd function. Because the relation holds for every function, it is unlikely to be useful. Likewise every function is of the form $f(s) + g(s)$ where $f(s) = f(1-s)$ and $g(s) = -g(1-s)$. I see those facts as in the same category as: every function is of the form $g(s) + g(s+1)$. Or $g(s) + g(s+a)$ for whatever $a$ you choose. Charming but not useful. | |
| Aug 29, 2022 at 11:09 | comment | added | Sylvain JULIEN | The fact that your operator $T$ is RH preserving is quite interesting. Has it any fixed points? Moreover, does it commute to any automorphism of an L-rig (see my questions on this website using L-rig as a keyword)? | |
| Aug 29, 2022 at 6:49 | comment | added | Dan Romik | @SalvoTringali fixed, thanks. | |
| Aug 29, 2022 at 6:48 | history | edited | Dan Romik | CC BY-SA 4.0 |
added 79 characters in body
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| Aug 29, 2022 at 4:48 | comment | added | Salvo Tringali | @DanRomik There is a typo in the 3rd displayed equation: $n^{s-1}$ should be $n^s$ right before the "${}= \zeta(s)$". It's nothing important, but I thought you may want to fix it. | |
| Aug 29, 2022 at 3:14 | comment | added | Dan Romik | @DavidFarmer I do enjoy messing around with cool formulas whether they are useful or not. But as for your “not useful” remark, surely as a general principle it’s not always true that explicit formulas are a distraction and don’t add anything useful as compared to general/more abstract ways of looking at things? In other words, I’m not sure I share your intuition on the usefulness issue, though you may be right of course. | |
| Aug 29, 2022 at 3:11 | comment | added | Dan Romik | @DavidFarmer property 1 of which question? Are you seeing a link somewhere to another question? (This one, perhaps?) | |
| Aug 29, 2022 at 2:27 | comment | added | David Farmer | As noted in Property 1 of the original question, every function $f(s)$ is formally of the form $g(s) + g(s+1)$: just let $g(s) = f(s) - f(s+1) + f(s+2) - f(s+3) + ...$. If $f(s)$ is a Dirichlet series then one can check that the sum defining $g(s)$ converges. This suggests to me that the expression $\zeta(s) = F(s) + F(s+1)$ will not be useful. In particular, the fact that you have a formula for $F(s)$ is just a distraction. | |
| Aug 28, 2022 at 18:21 | history | edited | Dan Romik | CC BY-SA 4.0 |
improved the explanation of the reformulated functional equation
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| Aug 28, 2022 at 13:52 | history | edited | YCor |
edited tags
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| Aug 28, 2022 at 13:50 | answer | added | Gui | timeline score: 1 | |
| Aug 26, 2022 at 7:51 | history | edited | Daniele Tampieri | CC BY-SA 4.0 |
Minor Math Jaxing (formula hyperlinking + other edits)
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| Aug 26, 2022 at 6:25 | history | asked | Dan Romik | CC BY-SA 4.0 |