A short note on large numbers

It is not very often that we encounter large numbers in mathematics; in fact, in some fields it’s sometimes quite common not to encounter numbers at all. Most numbers you meet are relatively benign and harmless: 0, 1, 2, 3, e, π; you don’t need much apart from those. In any case, even to an aspiring toddler, they are not large.
But sometimes there naturally appears a number that is obscenely large. Well, it depends on what you mean by naturally, I suppose, for there are so many possible questions to ask, some of them must give large results. But as far as I have seen from my studies, usually when you ask nice questions, you get nice answers (well, if you get answers at all). As an example, the infinite sum \sum_{n=1}^{\infty} \frac{1}{n}  diverges, but the infinite sum \sum_{n=1}^{\infty} \frac{1}{n^2}  converges to \frac{\pi^2}{6} . Somewhere along the way, you can find a value of α such that the sum \sum_{n=1}^{\infty} \frac{1}{n^{\alpha}}  converges to any number you want, but that would be picking a rather specific target, and the problem sounds contrived. Asking about n and n2 seems very natural, though.
But today I ran into the prime counting function – π(x) – which counts how many primes there are from 1 to x. It’s not that easy to calculate, but we can use approximations. A relatively-ok one is called li, for logarithmic integral, defined as

li(x) = \int_{0}^{x} \dfrac{dt}{\ln t} .

It’s not an entirely bad approximation: here you can see the number of primes, and the li(x) function in the same graph:

num_of_primes

It may seem by looking at the data that li(x) is always larger than π(x), but this is not the case. In fact, the quantity li(x) – π(x) changes sign infinitely often (proven by Littlewood). At what value does the first sign change happen? Meaning, what is the first x such that that π(x) > li(x)?
We don’t know, and whenever we don’t know the exact answer, we try to bound it from above and from below. Lower bounds are found by computing both functions for increasing x values, while upper bounds require theoretical results.
Currently, we know that the first sign change happens at

10^{14} < x < e^{727.95135}

I find this surprising.
First, the current upper bound is enormously large. There is always room to wonder when we encounter numbers in proofs that are larger than the number of atoms in the universe. It may be just an artifact of the proof, the real number may be much smaller, but for me, it certainly causes eyebrow movement.
Second, the lower bound isn’t miniscule either. For a phenomenon that happens infinitely often, it sure is taking its time. Many interesting prime properties can be found early on in the prime sequence (I suppose that’s how we bothered looking for them in the first place), but this one can not. You may say, of course, that the question isn’t that natural – after all, li(x) is just another approximation function – but as an approximation, its relative simplicity and straightforwardness are quite attractive.
Primes are rooted deep in number theory; if we wish to understand mathematics, we must understand primes.
And this causes me to ponder:
Mathematics does not care about our lovely intuition of what constitutes a “big number”. It laughs in the face of the adorable humans, trying to understand the world within their own references and scales. There is an infinite amount of natural numbers. Period. There is an infinite amount of primes. Period. Name a number as large as you like, but once you name it, know that it is insignificant and tiny when compared to the vastness and infinity of those that follow it. It is hard to comprehend just how many natural numbers there are; perhaps the real surprise should be the sheer amount of results we have that do not involve incredibly large numbers.
Don’t you find that reassuring?

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