What are these mysterious beasts?

Well, π(x) is a function that tells us how many prime numbers there are less than or equal to x. For example, if we want to know how many primes there are below 10 (which would include 2, 3, and 5), we can use the formula:

π(10) = 4

So that’s pretty straightforward. But what about li(x)? That one might be a bit more confusing at first glance. It stands for “logarithmic integral” (which is basically just a fancy way of saying “the antiderivative of the log function”), and it looks like this:

li(x) = _2^x (1/ln t) dt

Now, if you’re not familiar with calculus or integrals, that might look pretty intimidating. And it turns out that this approximation gets better and better as x grows larger (which makes sense, since the more primes you have, the closer your estimate will be).

So what’s the big deal? Well, for a long time mathematicians were trying to figure out how close li(x) actually is to π(x), and whether there was some kind of explicit upper bound that we could use to measure their difference. And in 2016, Trudgian finally cracked the case!

Here’s what he came up with:

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π(x) li(x)

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0.2795 * x / (log x)^3/4 * exp(-sqrt(log x / 6.455))

Now, if you’re not a math whiz, that might look pretty intimidating at first glance. But let me break it down for you: this formula tells us that the difference between π(x) and li(x) is always less than or equal to some number (which we can call “the upper bound”) that depends on x. And as x gets larger, this upper bound gets smaller and smaller which means that our approximation of pi using li becomes more and more accurate!

Pretty cool stuff, right?