Instead, we’re going to break it down into simple terms so even your grandma can understand it!
First things first what is the prime number theorem? Well, in short, it says that if you take a really big number and divide it by the total number of primes less than or equal to that number (let’s call this “P”), then the result will be pretty close to 1. For example, let’s say we want to find out how many prime numbers there are between 1 and 100. To do this, we can use a handy dandy formula called the Mertens function (which is basically just a fancy way of saying “counting primes”).
So if we plug in our values for n=100 and k=2 (because we want to count prime numbers up to 100), we get:
M(100, 2) = (-1)^2 * ln(100^(-2)) / (2 * pi) * gamma(3)
Where “ln” is the natural logarithm function and “gamma” is a special mathematical constant. Anyway, if you plug in those values into our formula, you get:
M(100, 2) = (-1)^2 * ln(100^(-2)) / (2 * pi) * gamma(3)
Which simplifies to:
M(100, 2) = -ln(100)/(4*pi)*gamma(3)
And if you calculate the value of this function using a calculator or computer program, you’ll get something like:
-0.0069875…
So according to our formula, there should be around 69 primes between 1 and 100 (which is pretty close to the actual answer of 66). But how do we know this is true? Well, that’s where formalizing an analytic proof comes in! Essentially, what we want to do is take our formula for M(n, k) and show that it converges to a certain value as n gets really big. And the way we do this is by using some fancy math tricks (like integrals and limits) to prove that:
lim_n->infinity [M(n, k)] = 1/k * ln(n) gamma(0)/gamma(k+1) + sum_{j=1}^infty (-1)^j*B_j*(ln(n))^j / (j! * j^k)
Where “lim” is the mathematical symbol for “as n approaches infinity”, and “sum” represents a series of terms that get smaller and smaller as we go along. And if you’re wondering what all those other symbols mean, well…that’s where things start to get a little bit more complicated!
And if you ever find yourself lost in all those fancy math equations, just remember: sometimes less is more (especially when it comes to understanding complex concepts)!