Timeline for Can one show that $|\zeta'(x) / \zeta^2(x)| \leq 1/(x-.5)$ for $x\in\mathbb{R}\cap [1,\infty)$?
Current License: CC BY-SA 4.0
27 events
| when toggle format | what | by | license | comment | |
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| Nov 3 at 9:55 | vote | accept | Haidara | ||
| Nov 3 at 9:48 | vote | accept | Haidara | ||
| Nov 3 at 9:54 | |||||
| Nov 3 at 5:19 | answer | added | GH from MO | timeline score: 6 | |
| Nov 2 at 22:33 | history | edited | Mark Schultz-Wu | CC BY-SA 4.0 |
more informative title
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| Nov 2 at 20:47 | comment | added | GH from MO | Please use a high-level tag like "ca.classical-analysis-and-odes". I added this tag now. Regarding high-level tags, see meta.mathoverflow.net/q/1075 | |
| Nov 2 at 20:46 | history | became hot network question | |||
| Nov 2 at 20:45 | history | edited | GH from MO |
edited tags
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| Nov 2 at 20:16 | vote | accept | Haidara | ||
| Nov 2 at 20:16 | |||||
| Nov 2 at 19:41 | vote | accept | Haidara | ||
| Nov 2 at 20:16 | |||||
| Nov 2 at 19:04 | comment | added | Haidara | @Conrad I agree with you that it might not be surprising but the hardness of the question lies in the pole at x=1 because it is not possible to bound a function near a pole with just finite numbers. Also I am just a high schooler so I might be wrong :). Also I see you have answersd some questions about the Riemann zeta function before so it seems that you have experience in the field so I hope you can answer my question. | |
| Nov 2 at 18:58 | comment | added | Haidara | @Conard first it is surprising for me because it does not break eventually like other guesses and also I want to prove the inequality holds for all x>1 without excluding any small interval because for lets say for x>2 I have ideas for the proof using the first two terms of the dirichlet series of each function together with the value of that function but I can't extend it for the more general case. | |
| Nov 2 at 18:16 | comment | added | Conrad | why it is suprising? Note that $-x\zeta'(x)$ is quite small (already for $x=10$ it is less than $7/1000$ while $\zeta^2(x)$ is about $1$ while $\frac{x}{x-1/2}$ is also close to $1$ so except on a small interval say $[1,10]$ just very coarse inequalities will do the job | |
| Nov 2 at 16:24 | comment | added | Haidara | @PeterMueller Yes, I see. | |
| Nov 2 at 16:23 | comment | added | Haidara | @SidharthGhoshal Nice approach! Can you try if it does the job? | |
| Nov 2 at 16:15 | comment | added | Sidharth Ghoshal | Both functions have a globally converging Laurent series representation at $x=1$. It might be fruitful to then try to prove “is the absolute value of the coefficient $c_n$ of $(x-1)^n$ in the $\zeta’$ Laurent series eventually always $\le$ the coefficient $c_{n+1}$ of the $\zeta^2$ Laurent series at $(x-1)^{n+1}?$”. if yes then @PeterMueller’s result is true. Now this becomes a question of how well can we analyze the asymptotics of the stieltjes constants. | |
| Nov 2 at 15:35 | comment | added | Peter Mueller | I guess the stronger inequality $\lvert\frac{\zeta'(x)}{\zeta^2(x)}\rvert\leq \frac{1}{x}$ holds too. | |
| Nov 2 at 11:37 | comment | added | Jannik Pitt | @AlekseiKulikov In mathematics, not every heuristic reasoning counts as a proof. Simply checking an inequality for finitely points with a computer with finite precision will always never be a proof! Also, there is more to mathematical results than simply being true or false -- we want to know why they are true (or false!). However, I make no claims that this question here is of particular interest, which I cannot judge. | |
| Nov 2 at 10:58 | comment | added | Haidara | @AlekseiKulikov also we are using computers to verify the riemann hypothesis and it has been verified upto a very big number but that doesn't mean we can believe the RH without a rigorous mathematical proof so we need to research even if the result has been verified by computers for very large numbers. | |
| Nov 2 at 10:50 | comment | added | Haidara | @AlekseiKulikov in your case the computer can give us the proof since it is based on some algorithm that can do all the computation needed to be sure that the result is true or false but my question is about an infinite continuous range and this function is highly sensitive to precision errors that's why it should be proven before we can believe that it's true in general. | |
| Nov 2 at 10:31 | comment | added | Aleksei Kulikov | @StevenLandsburg in the same way as writing two thousand-digit numbers $A, B$ and a two-thousand-digit number $C$ and asking if $A*B>C$ is not a research level question, even though I will not be able to solve it without a computer. | |
| Nov 2 at 9:46 | comment | added | Haidara | @StevenLandsburg I am already doing research about a more general result but this is a simplification of my original problem that I couldn't prove myself. I hope someone can help me here since I am not an academic student. | |
| Nov 2 at 9:37 | comment | added | Steven Landsburg | @AlekseiKulikov : If for some range this is true only "by a computer", then how can it not be a research question? | |
| Nov 2 at 9:13 | comment | added | Haidara | Do you have any ideas of what a proof should look like? | |
| Nov 2 at 9:06 | review | Close votes | |||
| Nov 9 at 11:11 | |||||
| Nov 2 at 8:26 | comment | added | Aleksei Kulikov | For $x$ close to $1$ this is true by the Taylor expansion, for $x$ close to $+\infty$ this is true by the Dirichlet series expansion, for the intermediate $x$ this is true by a computer. But this is not a research level question. | |
| S Nov 2 at 8:14 | review | First questions | |||
| Nov 2 at 8:51 | |||||
| S Nov 2 at 8:14 | history | asked | Haidara | CC BY-SA 4.0 |