As all analytic number theorists know, iterated logarithms ($\log x$, $\log \log x$, $\log \log \log x$, etc.) are prevalent in analytic number theory. One can give countless examples of this phenomenon. My question is, can someone give an intuitive account for why this is so? Specifics regarding any of the famous theorems involving iterated logarithms are welcome. Many thanks!

EDIT: Thank you so much for the answers so far! I'm still trying to get a better intuition on how a $\log \log \log$ or $\log \log \log \log$ arises, especially in Littlewood's 1914 proof that $\pi(x)-\operatorname{li}(x) = \Omega_{\pm} \left(\frac{\sqrt{x}\log \log \log x}{\log x}\right) \ (x \to \infty)$ or Montgomery's conjecture that $\limsup_{x \to \infty}\dfrac{\lvert\pi(x)-\operatorname{li}(x)\rvert}{\;\frac{\sqrt{x}\, (\log \log \log x)^2}{\log x}\;}$ is finite and postive. I admit to knowing nothing (yet) about sieve theory, so I will have to dive into the proof of the prime gap theorem by Tao, Maynard, et. al. Can someone give a more precise account of how the $\log \log \log$ or $\log \log \log\log$ arises in the proof? I'm very familiar with why occurrences of $\log \log$ happen, but once you get to $\log \log \log$, I'm still a bit mystified. Also, is there a good introduction to sieve theory where I could start, or should I just dive right in to the papers on large prime gaps?

FURTHER EDIT: Can someone also explain intuitively the reason for the $\log \log \log$ in Littlewood's theorem? Historically, was this the first occurrence of a triple log in number theory?

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    $\begingroup$ My guess is that apart from the PNT, a "metareason" for this would be the validity of probabilistic Ansätze (esp. en.wikipedia.org/wiki/Cram%C3%A9r%27s_conjecture). In such settings double logarithms are common, as in e.g. en.wikipedia.org/wiki/Law_of_the_iterated_logarithm and en.wikipedia.org/wiki/…. $\endgroup$ Nov 15 at 20:58
  • $\begingroup$ I never understood how that law applies, since it works only outside a set of measure zero. But in some cases in number theory, as in the study of prime gaps, there is a $\log \log \log \log$ that occurs. I don't think the law of the iterated logarithm is enough to explain this. And nothing I've seen in analytic number theory really uses the law. If you use the law in trying to study the Mertens function, for example, you probably get the wrong order of growth. (There are competing conjectures, and the one with a double log is less likely to be true than the one with the triple log.) $\endgroup$ Nov 15 at 21:26
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    $\begingroup$ @Jesse Elliott You may find interesting Chapter 1 of Hall and Tenenbaum's "Divisors." $\endgroup$ Nov 16 at 1:18
  • $\begingroup$ Regarding the final question in your edit: I would recommend the latter half of Dimitris Koukoulopoulos' "The Distribution of Prime Numbers" as a very accessible introduction to sieve theory. (It is probably my favorite mathematical text.) In particular, the last section (Part 6) covers the results on long and short gaps that you are interested in. $\endgroup$ Nov 17 at 1:09
  • $\begingroup$ (The most comprehensive account of sieve methods is Friedlander and Iwaniec's "Opera de Cribro." But, being written in 2010, it does not include the state-of-the art work on small and large gaps. These notes by Kevin Ford are also nice faculty.math.illinois.edu/~ford/sieve2020.pdf, and do cover small and large gaps.) $\endgroup$ Nov 17 at 1:19

There are two main sources of repeated logs. (These sources can be further refined into natural subcategories, but I'll only mention a couple of those subcategories.) Those two main sources are:

Type 1: Repeated logs occur because that is just the truth of the matter.

One of my favorite examples is a 2008 theorem of Kevin Ford, solving the multiplication table problem. The theorem states that $$ |\{a\cdot b\, : a,b\in \{1,2,\ldots,N\}\}|\asymp \frac{N^2}{\log(N)^c(\log\log(N))^{3/2}}, $$ where $c=1-\frac{1+\log\log(2)}{\log(2)}$.

Lest you believe that the $(\log\log(N))^{3/2}$ factor is a consequence of this being a 2-dimensional problem, it also shows up in the other dimensions. See this other question for more information.

In some cases it is much easier to see where these extra log's come from. For instance, when turning sums over integers into sums over primes, this often leads to an extra log coming into force, just from the nature of the problem at hand and asymptotics with primes.

For instance, we have $$ \sum_{n=1}^{N}\frac{1}{n}=\log(N)+\gamma+o(1)\ \text{ while }\ \sum_{p\leq N,\ p\text{ prime}}\frac{1}{p}=\log\log(N)+B+o(1), $$ where $\gamma$ and $B$ are well-known constants. These two asymptotics can be thought of as discrete version of the integral equalities $$ \int\frac{1}{x}\, dx = \log(x) \ \text{ while }\ \int\frac{1}{x\log(x)}\, dx=\log\log(x). $$

Since primes occur all over number theory, and they also come weighted with an extra log factor, this often contributes extra double-log factors.

Type 2: Repeated logs occur as an artifact of our current best machinery.

For example, Rankin showed in 1938 that the largest prime gap below $N$, for $N\gg 0$, is at least $$ \frac{1}{3}\frac{\log(N)\log\log(N)\log\log\log\log(N)}{(\log\log\log(N))^2}. $$ These extra logs happen when optimizing inequalities, and when using the known machinery of the day. But they do not represent a fundamental truth about the problem. The constant $\frac{1}{3}$ has been slowly improved. Recently, in 2014, Ford, Green, Konyagin, and Tao improved this bound, and Maynard also did so independently, by replacing the fraction $\frac{1}{3}$ by an arbitrary number. See this preprint and this other preprint for more details. Later, these five mathematicians together removed the square from the denominator.

If you read the proofs, the logs are coming from the current state of the art sieve methods, together with bounding techniques. When you solve for the best fit functions to undo some of the exponentiation that occurs in calculations, the logs just fall out.

In these types of problems, it is not inconceivable (and actually occurs quite regularly) that one new idea is applied to the problem, and the asymptotic changes (sometimes involving more multi-log factors, to account for the small additional room for improvement that was gained). What is surprising about Rankin's bound is that even though it is far from the predicted asymptotic, each extra idea only changed the constant out front---at least until recently.

Edited to add: Working through a well-written proof will, of course, give a deeper understanding of where iterated logs arise in the problem at hand. That is certainly true for the three examples above.

However, if you are not yet familiar with sieve theory, or the circle method, I wouldn't recommend working through the big proofs of those theorems mentioned above and in your question (at least, not initially). Rather, I would recommend starting with an introductory text on sieves, such as Cojocaru and Murty's book "An Introduction to Sieve Methods and their Applications". Double-logs occur almost at the very beginning. Triple-logs show up in the exercises in Chapter 5 (and perhaps earlier). Indeed, problem 25 is a typical example of how a triple log is introduced to improve an asymptotic.

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    $\begingroup$ $+1$, but this seems much more like a comment than an answer $\endgroup$ Nov 16 at 6:38
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    $\begingroup$ @mathworker21: How would it fit in the Comments section, then? =) $\endgroup$ Nov 16 at 7:09
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    $\begingroup$ @mathworker21 That's a strange thing to say about the only answer that gets to the heart of the matter. $\endgroup$
    – Will Sawin
    Nov 16 at 19:03
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    $\begingroup$ @JesseElliott I edited my post to answer your new questions. I personally found Cojocaru and Murty's book very enlightening. I really liked chapter 5, in particular. $\endgroup$ Nov 16 at 21:48
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    $\begingroup$ @mathworker21 How is this not an answer to the question? $\endgroup$ Nov 18 at 4:16

I wouldn't say that there is a single all-prevailing reason, but here are some easily discernible sources:

  • Given a Dirichlet series $$\sum_{n=1}^{\infty}\frac{a(n)}{n^s}$$ in its abscissa of absolute convergence, we have that $$\frac{d}{ds}\sum_{n=1}^{\infty}\frac{a(n)}{n^s}= -\sum_{n=1}^{\infty}\frac{a(n)\log n}{n^s}$$
  • The function $\Gamma(s)$ is ubiquitous in analytic number theory (one need not look beyond the functional equation of $\zeta(s)$), and $$\frac{\Gamma'}{\Gamma}(z) = \log z -\frac{1}{2z}+ O(z^{-2}).$$
  • Mellin inversion, which allows us to express partial sums of arithmetic functions in terms of contour integration, is (essentially) a logarithmic change of variables away from Fourier inversion.

This does not account for everything (these are largely motivated by multiplicative number theory and connections with $L$-functions), but it accounts for a lot, including the asymptotic

$$\sum_{p\leq x}\frac{1}{p} = \log\log x + B + O((\log x)^{-1}).$$

This single asymptotic accounts for a lot of the presence of $\log x$ and its higher compositions in sieve theory. Combine these with ideas like partial summation, and the logs accumulate.

  • $\begingroup$ Thank you! Can you explain a little more about how the logs accumulate? For example, where does a $\log \log \log$ or $\log \log \log \log$ come from? Is there some kind of iteration involved? How is partial summation involved? $\endgroup$ Nov 16 at 21:15
  • $\begingroup$ @JesseElliott That may follow from the processes in which logs get plugged into some other logs $\endgroup$
    – TravorLZH
    Nov 17 at 0:58
  • $\begingroup$ @TravorLZH That's too vague and not very informative for me. That's exactly what an iterated log is. $\endgroup$ Nov 17 at 11:37

One can often attribute this phenomenon to Mertens' product formula:

$$ \prod_{p\le x}\left(1-\frac1p\right)\sim{e^{-\gamma}\over\log x} $$

Using this fact, we can deduce a minimal order for Euler's totient. That is

$$ \liminf_{n\to\infty}{\varphi(n)\log\log n\over n}=e^{-\gamma} $$

Using similar tricks, one can obtain a maximal order for divisor sum function:

$$ \limsup_{n\to\infty}{\sigma(n)\over n\log\log n}=e^\gamma $$

Proofs of these results are available in Hardy & Wright's An Introduction to the Theory of Numbers and Tenenbaum's Introduction to Analytic and Probabilistic Number Theory.


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