Instead, let me break it down for you in a way that even my grandma could understand!
Well, if you’re not already familiar with this concept (in which case I suggest taking some remedial math classes), a prime number is any positive integer greater than 1 that has no positive divisors other than 1 and itself. So for example, 2, 3, 5, 7, 11, 13, 17, etc., are all prime numbers.
Now the PNT this is a theorem in number theory that describes how many primes there are up to a given limit (or “cutoff”). Specifically, it states that if you take any large enough number n and count the number of primes less than or equal to n, then the ratio of this number to n itself approaches 1/ln(n) as n gets larger.
So for example, let’s say we want to find out how many prime numbers there are between 2 and 50 (inclusive). Well, if you do some quick counting, you’ll see that the answer is: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, and 37. That makes a total of 25 primes in this range but what’s more interesting is the ratio between these numbers:
(Number of Primes / Total Number) = (25/50) = 0.5
Now let’s take that same calculation, but instead of using our fingers and toes to count up all those primes, we can use a fancy math function called the Li function (or “L” for short). This is essentially just a way of calculating the number of prime numbers less than or equal to n in a more efficient manner.
So if you plug 50 into our trusty L function, you’ll get an answer that looks something like this:
L(x) = x * (ln(x) + γ ln(ln(x)) 1/2*ln^2(ln(x))) + O(x/(ln(x))^3)
where “O” stands for “big-oh notation”, which is a fancy way of saying that this term gets smaller and smaller as x gets larger. And if you plug in our cutoff value of 50, then the answer comes out to be:
L(50) = 142913867 (rounded to nearest integer)
Now let’s compare that number with what we got from our previous calculation using good old-fashioned counting:
Number of Primes / Total Number = (25/50) = 0.5
So if you divide the L function result by the total number, then you get an answer that looks something like this:
(L(x) / x) * (ln(x))^(-1) + O(x/(ln(x))^3) / x * (ln(x))^(-1) = 0.596…
And if you compare that to the PNT, which says that this ratio should approach 1/ln(n), then you’ll see that they match up pretty closely! In fact, for large enough values of n, these two numbers are essentially indistinguishable from each other.
And if you’re still feeling confused or overwhelmed by all this math talk, just remember: at least you’re not my grandma!