1. It’s a long-term goal of mine to understand the proof of the Prime Number Theorem. The PNT, for those who are unfamiliar, is that the number of primes less than or equal to N is approximately N/ln(N), with the approximation improving as N grows to infinity; the percent error of the approximation approaches zero as N grows. I found a really interesting paper that walks through a proof. (The paper won an award for math exposition.) I’ve only done a quick first read-through, but it seems interesting. On that first pass, I think I now get why we even bother with the von Mangoldt symbol. Baby steps. If anyone would like to read along with me, y’all are welcome to.

One question I have is whether it’s mathematically impossible to find a function that exactly equals the number of primes less than or equal to a number. The N/ln(N) approximation is good, but it’s just an approximation with a known rate of error — and while the percent error drops to zero, the absolute error grows without bound. Is there a proof that a function which counts the exact number of primes less than N is impossible?

(I could be more precise about this. Obviously the function which sums the number 1 over all primes p less than or equal to x is an exact function. But it’s obviously not what I’m looking for. There’s probably a formal way to describe exactly what I’m looking for it. It’s probably either “a function containing only a certain set of symbols” or “a function of at most O(log n)” or something.)

2. If someone gives you the equation “ax = b”, you can solve that really easily with elementary algebra; the solution is x = b/a. Likewise if someone gives you an equation involving x-squared; the solution is the quadratic formula. The solutions get more and more complicated for x-cubed and the fourth power of x. There’s a theorem (Abel-Ruffini) which says that no solution using only addition, subtraction, multiplication, division, and root extraction (square roots, cube roots, etc.) is possible for the fifth power of x and higher in the general case. When I limit it to “the general case”, I mean that certain specific equations at higher powers may have solutions: x100-1=0 has the solution x=1. But the general equation ax5+bx4+cx3+dx2+ex+f=0 has no solution, nor do equations with higher powers.

The question I’ve had for a while (apart from knowing enough Galois theory to know how to prove Abel-Ruffini) is what happens if you allow mathematical operations other than addition, subtraction, and the rest. What if you allow sines and cosines? Fourier series and their inverses? Bessel functions? Elliptic functions? The Wiki says that a certain kind of transformation turns the general quintic into a simpler quintic polynomial, which can then be solved using a Bring radical. Is the Bring radical the simplest function which can solve the general quintic?

So then my first question is: what’s the simplest set of functions you can add to addition, subtraction, multiplication, and division, such that the general quintic has a solution in this augmented set?