First off, Littlewood’s proof from 1914. This guy was like the Einstein of primes (or maybe more accurately, the Newton of primes). He proved that if you take any given number n, there are infinitely many prime numbers less than or equal to n. That’s right, Infinite primes for the win!
But wait, what does “less than or equal to” even mean? Well, let me break it down for ya: if we have a set of numbers (let’s call them X), and we want to find all the prime numbers in that set that are less than or equal to some other number (let’s call it n), then we can write this as “X n”. So basically, Littlewood was saying that there are infinitely many primes in any given set of numbers up until a certain point (n).
Now, let’s fast forward to the year 2004. This is when Goldfeld came along and proved something even more mind-blowing: an elementary proof for the prime number theorem! That’s right, No fancy math or complicated formulas required just good old fashioned logic and reasoning.
So what exactly does this mean? Well, let me explain it to you in simple terms: if we have a set of numbers (let’s call them X), and we want to find out how many prime numbers are in that set up until some other number (let’s call it n), then the formula for calculating this is “X ÷ log(n)”. So basically, Goldfeld was saying that if you divide any given set of numbers by the natural logarithm of a certain point (n), you will get an approximation for how many prime numbers are in that set up until that point.
Now, some practical applications for these asymptotic laws. For example, imagine you have a list of 100 numbers and you want to find out how many primes there are in that list. Well, if we use Littlewood’s proof, we can say that since the set of all prime numbers is infinite (according to his theorem), then there must be infinitely many primes in any given subset of those numbers up until a certain point (let’s call it n). So basically, no matter how big or small your list is, you will always find an infinite number of primes within that set.
On the other hand, if we use Goldfeld’s proof to calculate the approximate number of prime numbers in our list up until a certain point (let’s call it n), then we can say that this value will be very close to the actual number of primes in that set. So basically, by using these asymptotic laws, we can make some pretty accurate predictions about how many prime numbers there are within any given subset of numbers up until a certain point (let’s call it n).