Timeline for Positivity of $L(1,\chi)$ for real Dirichlet's character
Current License: CC BY-SA 2.5
11 events
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| May 26, 2010 at 7:52 | comment | added | Wadim Zudilin | Junkie, thank you very much for the links to Chowla's method, which is not surprising to me from the analytic point of view, but I did not expect it working in this $L$-function context. It reminds me about Pr\'evost's proof of Ap\'ery's theorem (irrationality of $\zeta(2)$ and $\zeta(3)$). I am too thankful today, but I am very happy to see so many useful comments on the question. | |
| May 26, 2010 at 7:17 | comment | added | Junkie | It is not the same, but Chowla then Rosser chased around vaguely similar ideas (weighted sums) to show that $L(s,\chi)>0$ for real $s>0$ for various real $\chi$. Chowla: matwbn.icm.edu.pl/ksiazki/aa/aa1/aa119.pdf Rosser: ams.org/journals/bull/1949-55-10/S0002-9904-1949-09306-0/… | |
| May 26, 2010 at 2:22 | comment | added | KConrad | Wadim and David: there's an easier reason why L(1,chi) can't be negative for real nontrivial chi: the function L(s,chi) for s > 0 is obviously real-valued. As s --> infty it tends to 1, which is positive, and we know L(s,chi) is nonzero for s > 1 by the Euler product. Therefore by continuity L(s,chi) > 0 for s > 1, hence by taking a limit from the right L(1,chi) is definitely not negative. It's not 0 either (harder!), so L(1,chi) > 0. | |
| May 26, 2010 at 2:02 | vote | accept | Wadim Zudilin | ||
| May 26, 2010 at 1:56 | comment | added | Wadim Zudilin | I should have had in mind this implication. Very nice to start a day with nice knowledge. Thank you for this lesson, David! | |
| May 26, 2010 at 1:46 | comment | added | David Hansen | Oh, no, the inequality $L(1,\chi)>0$ for $\chi$ real goes back to Dirichlet, and is an immediate consequence of his famous class number formula. | |
| May 26, 2010 at 1:15 | comment | added | Wadim Zudilin | The story about Fekete polynomials is really intriguing. Even the only $\chi$ in this case is the Legendre (Jacobi) symbol, it seems that similar results should be true for real $\chi$ in general. As for my extra question, the fact that $g(x)$ alternates on $(0,1)$ many times does not imply that the "average" $L(1,\chi)$ of $g(x)/(1-x^m)$ could be done negative. So, are there examples with $L(1,\chi)<0$? | |
| May 26, 2010 at 1:01 | comment | added | David Hansen | fekete, sorry! I'm not sure I understand your second question. | |
| May 26, 2010 at 1:01 | history | edited | David Hansen | CC BY-SA 2.5 |
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| May 26, 2010 at 1:00 | comment | added | Wadim Zudilin | Thanks, David! Never heard of them... Feteke or Fekete? And is $L(1,\chi)>0$ for some $m$? | |
| May 26, 2010 at 0:52 | history | answered | David Hansen | CC BY-SA 2.5 |