Timeline for Normal numbers, Liouville function, and the Riemann Hypothesis
Current License: CC BY-SA 4.0
21 events
| when toggle format | what | by | license | comment | |
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| May 7, 2021 at 18:22 | comment | added | Kurisuto Asutora | Normality is essentially a form of the strong law of large numbers, requiring only $o(n)$ cancellation. A law of the iterated logarithm requires square-root cancellation near $O(\sqrt{n})$, which is much more to ask for. Very informally, an LIL requires a higher degree of "independence" than an SLLN. | |
| May 7, 2021 at 15:57 | history | edited | Vincent Granville | CC BY-SA 4.0 |
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| May 7, 2021 at 14:46 | comment | added | Vincent Granville | @Kurisuto: thank you, very useful reference. That means a sequence of (say) binary digits can represent a normal number even if it does not satisfy the law of the iterated logarithm, thus removing some doubts about Chowla's conjecture. | |
| May 7, 2021 at 8:20 | comment | added | Kurisuto Asutora | I think you don't need independence to deduce normality. You might find this reference interesting, on the connection of normality with digital dependencies more or less of the type you suggest here: arxiv.org/abs/1804.02844. | |
| May 6, 2021 at 3:38 | comment | added | Vincent Granville | Upon reading more material on this topic, it looks like the law of the iterated algorithm does not apply in this context; this is in alignment with your comment. In short, the Liouville numbers are not random enough, and I think it implies that $\lambda^*$ is not a normal number. Some is discussed in this paper: arxiv.org/ftp/arxiv/papers/1106/1106.1895.pdf. Though that paper has some deep flaws. | |
| May 3, 2021 at 20:33 | history | edited | Vincent Granville | CC BY-SA 4.0 |
Addition at the bottom of "Update 2": if some (explicitly specified) invariant distribution attached to $\lambda^*$ is uniform on $[0,1]$, it would also imply RH.
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| May 3, 2021 at 17:52 | vote | accept | Vincent Granville | ||
| May 3, 2021 at 8:38 | comment | added | GH from MO | I don't think that it is proved that Gonek's conjecture (or its analogue for the Liouville function) contradicts Chowla's conjecture. | |
| May 3, 2021 at 8:02 | comment | added | Vincent Granville | @GHf: If that constant $C$ in my post is zero, then $\lambda^*$ can't be a normal number, I think. Again, no implications for RH, but it seems to imply that in that case, Chowla's conjecture is wrong (someone pointed out that it is equivalent to $\lambda^*$ being normal). | |
| May 3, 2021 at 7:42 | comment | added | GH from MO | Gonek's conjecture suggest that $C$ in your third display is actually zero. The conjecture states that $\limsup|\mu(1)+\cdots +\mu(n)|/\sqrt{n}(\log\log\log n)^{5/4}$is finite and positive. | |
| May 3, 2021 at 7:31 | history | edited | Vincent Granville | CC BY-SA 4.0 |
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| May 3, 2021 at 6:36 | history | edited | Vincent Granville | CC BY-SA 4.0 |
See section "Update 2", clarifying what I mean by ergodicity.
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| May 3, 2021 at 4:25 | history | became hot network question | |||
| May 3, 2021 at 4:18 | history | edited | Vincent Granville | CC BY-SA 4.0 |
Added details about the law of the iterated algorithm applied to the sequence $\lambda(k)$
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| May 3, 2021 at 4:09 | history | edited | Vincent Granville | CC BY-SA 4.0 |
See "update" section at the bottom, about transcendence (or not) of $\lambda^*$. Of course, $\lambda^*$ is irrational.
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| May 3, 2021 at 2:17 | history | edited | Vincent Granville | CC BY-SA 4.0 |
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| May 2, 2021 at 22:02 | comment | added | Ofir Gorodetsky | (cont.) If I understand correctly, your random function $n \mapsto X_n$ is known in the literature as a 'random multiplicative function' and is closely related to Rademacher random multiplicative functions. Proving normality for $\frac{1}{2}(1+\sum X_k/2^k)$ is the same as showing that $\sum_{n \le x} \prod X_{n+a_i} = o(x)$ with probability 1, and this should be provable unconditionally (e.g. compute first two moments and show concentration of these sums). In certain senses, normalized sums of $X_n$ do not behave normally (see Adam Harper's works). | |
| May 2, 2021 at 21:59 | comment | added | GH from MO | @OfirGorodetsky: We used the same words "widely believed", independently! | |
| May 2, 2021 at 21:54 | comment | added | Ofir Gorodetsky | Base-2 normality of your $\lambda^{*}$ is equivalent to a famous conjecture of Chowla, saying that $\sum_{n \le x} \prod_{i=1}^{h} \lambda(n+a_i) = o(x)$ for every $h \ge 1$ and sequence $a_1 < a_2 < \ldots < a_h$. In the last few years there has been significant progress towards this conjecture. It remains open but is widely believed to be true, and the theoretical evidence keeps growing. A concrete reference is Matomäki, Radziwiłł and Tao, 'Sign patterns of the Liouville and Möbius functions', Forum Math. Sigma 4 (2016), Paper No. e14. (cont.) | |
| May 2, 2021 at 21:53 | answer | added | GH from MO | timeline score: 13 | |
| May 2, 2021 at 20:23 | history | asked | Vincent Granville | CC BY-SA 4.0 |