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Is there some accepted, more concise notation for expressions like $\log \log \log n$? I just noticed an arXiv posting that quotes the bound $$ \frac{\log X \log \log X \log \log \log \log X} { \log \log \log X} $$ and I wonder if there is room for some notational improvement in this domain. Clearly $\log^k n$ is not an option, as that seems entrenched in the literature to mean $(\log n)^k$.

If there is no accepted, more concise notation, perhaps the community could invent a notation.

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    $\begingroup$ Is this research level? $\endgroup$
    – Andreas Thom
    Commented Apr 10, 2015 at 12:41
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    $\begingroup$ @Andreas, where, other than in research papers on analytic number theory, is one likely to come across $\log\log\log\log x$? $\endgroup$
    – Gerry Myerson
    Commented Apr 10, 2015 at 12:54
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    $\begingroup$ @GerryMyerson: You are right, with four logs it must be research. $\endgroup$
    – Andreas Thom
    Commented Apr 10, 2015 at 13:12
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    $\begingroup$ ... meaning, four logs is for lumberjacks, real research has infinitely many. $\endgroup$
    – Emil Jeřábek
    Commented Apr 10, 2015 at 15:14
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    $\begingroup$ I really don't understand why this question has been placed on hold. If asking for the notation for the all-ones vector is a valid question for MO, I don't see why this isn't. $\endgroup$
    – David Zhang
    Commented Apr 13, 2015 at 6:15

4 Answers 4

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Analytic Number Theorists have been using $\log_nx$ for the $n$-times iterated (natural) logarithm of $x$ for some time. See, e.g., the first page of this paper by Ford, Green, Konyagin, and Tao. Alternatively, Claudia Spiro used $L_4x$ for $\log\log\log\log x$ in this paper.

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    $\begingroup$ Ha! This is pretty much the same paper Joseph cites in the question --- just go past the abstract, and look at page 2! $\endgroup$
    – Gerry Myerson
    Commented Apr 10, 2015 at 12:40
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    $\begingroup$ Well -- ${\rm log}_n x$ could be mistaken for the logarithm for base $n$ of $x$. $\endgroup$
    – Stefan Kohl ♦
    Commented Apr 10, 2015 at 12:51
  • $\begingroup$ @Stefan, yes --- so whenever it's used for the iterated logarithm, the author(s) carefully explain just what they mean by it (as in the Ford et al. paper). $\endgroup$
    – Gerry Myerson
    Commented Apr 10, 2015 at 12:53
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    $\begingroup$ Excuse the rant, but just to go out of the way of the imbeciles who write $\log^nx$ to save two keystrokes (counting the space that TeX requires) from the completely unambiguous $(\ln x)^n$, ANT people go abuse a notation that half of humanity uses for something quite different! At the very least they could have come up with something like $\log_{(n)}x$ or $\log^{(n)}x$ (yes, that is $4$ more keystrokes, unless you define a macro). When will we mathematicians grow up? $\endgroup$
    – Marc van Leeuwen
    Commented Apr 10, 2015 at 13:17
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    $\begingroup$ @Marc, I think the part of humanity that has a use for logarithms-to-the-base-4 is disjoint from the part of humanity that has a use for $\log\log\log\log x$, so using $\log_4x$ for the latter isn't likely to even come to the attention of anyone who would have expected the former. And, as I mentioned, the convention is to explain the notation immediately upon first using it, so no one will be confused for long. $\endgroup$
    – Gerry Myerson
    Commented Apr 10, 2015 at 13:27
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I have seen the notation $F^{\circ2}(x)$ and $F^{\circ n}(x)$ in a Dynamical Systems book by Shlomo Sternberg (see e.g. section 2.2.2 Period doubling, there). Clearly $\circ$ indicates that we are talking about composition of functions and distinguishes it from $F^2(x)$ and $F^n(x)$ if you interpret the latter as $(F(x))^2$ and $(F(x))^n$. So one might perhaps use $\log^{\circ4}n$ and $\log^{\circ k}n$. I would personally prefer this, since $\log_4n$ could be confused with $\log$ with base $4$, though I am sure people in analytic number theory know what they mean with their choice of notation.

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  • $\begingroup$ As I have mentioned (twice!), people in analytic number theory explain what they mean with their choice of notation. $\endgroup$
    – Gerry Myerson
    Commented Apr 10, 2015 at 23:17
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    $\begingroup$ yes @GerryMyerson but everybody writes, nobody reads ... I did eventually read it (before you reminded me but after I posted my answer). Yet, I would argue that good notation needs little or no explanation. $\endgroup$
    – Mirko
    Commented Apr 10, 2015 at 23:28
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I don't know about generally accepted, but $\log^{(n)}X$ seems pretty concise. It conforms to the usual notation for functional iterations and there is no room for confusion with either taking powers or using different bases, e.g. $(\log_2^{(n)}X)^k$.

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    $\begingroup$ There is room for confusion for the $n$th derivative of the $\log$ function, though. $\endgroup$
    – Benoît Kloeckner
    Commented Apr 10, 2015 at 13:48
  • $\begingroup$ True. However, if it comes to the $n$th derivative of $\log$, the context is usually such that it's more preferable to use differential operator notation instead, or the simple explicit expression for it. $\endgroup$
    – M.G.
    Commented Apr 10, 2015 at 14:01
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    $\begingroup$ The slightly different notation $\log^{[n]}{X}$, which avoids clearly the conflict with derivatives, was my immediate reaction to the title of this question. $\endgroup$
    – Vesselin Dimitrov
    Commented Apr 11, 2015 at 1:00
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I find $(\text{log}^k)(n)$ quite clear, with little or no potential for misunderstanding.

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    $\begingroup$ To carefully specify composition, we can use $\log^{\circ k} n$ $\endgroup$
    – Gerald Edgar
    Commented Apr 10, 2015 at 13:40

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