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On https://en.wikipedia.org/wiki/Liouville_function there is written $L(n)>0.06 \sqrt n$ and $L(n)<-1.39 \sqrt(n)$ for infinitely many $n$.

On http://mathworld.wolfram.com/LiouvilleFunction.html they say it is unknown if $L(n)$ changes sign infinitely many times.

But those statements are contradicting each other. (so one has to contact that site which is wrong) So which one of the statements is true?

Hagen von Eitzen
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Stijn Cambie
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    [citation needed] – Hagen von Eitzen Apr 01 '16 at 12:39
  • As a general rule it seems to me that wikipedia is much more reliable on mathematical issues than mathworld. You might note that the wikipedia article includes a reference... – David C. Ullrich Apr 01 '16 at 12:39
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    @DavidC.Ullrich but seemingly not for the -1.39 claim – Hagen von Eitzen Apr 01 '16 at 12:40
  • @HagenvonEitzen Unless it does. The title of the reference for the 1.69 part is "Sign Changes in Sums of the Liouville Function" - certainly seems possible that the paper talks about sign changes... worth looking anyway. – David C. Ullrich Apr 01 '16 at 12:43
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    Mathworld also gives a reference, namely "Tanaka 1980". Then it is clear that this was the status in $1980$. – Dietrich Burde Apr 01 '16 at 12:43
  • @DavidC.Ullrich Good point. I merely checked which parts of the claims were formally covered by citations according to the footnote positions, not by references themselves. - I also just noticed that the German WP says "unknown", so is as behind as MW is (if the problem has really been settled). - At the same time, the French quotes the same Borwein et al. for the $>0.06\sqrt n$ part and says that the sign change problem is unsolved (with a reference to MW). – Hagen von Eitzen Apr 01 '16 at 12:49
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    Thanks to @tilper I just scanned through the Borwein et al. pdf. Apparently, the $0.06\sqrt n$ part appears as Theroem 2 there whereas the constant -1.3892783 is not contained in the text ... – Hagen von Eitzen Apr 01 '16 at 12:57
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    Googling -1.3892783 turns up many WP clones - and http://math.stackexchange.com/questions/1462068/liouville-function-sum-and-upper-density Meanwhile I did what I suggested in my first comment and flagged the Wikipedia claim – Hagen von Eitzen Apr 01 '16 at 13:16

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I found no evidence that $L(n) < 0$ for infinitely many $n$. But maybe I just missed it. I checked the relevant reference, which can be found here as of the time I write this - http://www.ams.org/journals/mcom/2008-77-263/S0025-5718-08-02036-X/

Click on "Full-text PDF" to read it.

Near the top of page 1685 it says, "...With this information, the method of Anderson and Stark shows that $$ L(n) > 0.027536\sqrt{n}$$ infinitely often."

Near the middle-bottom of page 1693 (just before the acknowledgments) it says, "With this computation, we verified that indeed $L(n) < 0$ for $10^9 \le n \le 2 \cdot 10^{14}$." I think that's the best they did in terms of when $L(n) < 0$.

  • Tanaka (1980) proved that $L(n)-{\sqrt{n}\over\zeta({1\over 2})}$ changes sign infinitely often, where $\zeta({1\over 2})=-0.684765236\ldots,$ hence $L(n) < 0$ infinitely often. The proof is mentioned here and appears here. – r.e.s. Oct 03 '19 at 13:58