Motivating the Riemann Hypothesis

This is a short introduction to the Riemann hypothesis and associated technical background for college math majors or advanced high school students. I use it to chronicle my own (limited) understanding of this important unsolved problem, because I am probably not going to be able to fit proper courses on analytic number theory or advanced complex analysis into my college coursework.

Short Version

We care about prime numbers because many questions about natural numbers can be easily answered using their prime factorization. The obvious first question about primes is about how they're distributed; in particular, how many there are up to some natural number \(x\). Various functions -- the most obvious one being \( x \over \ln x \) -- seem to approximate this "prime counting function", which we call \( \pi (x) \). The key idea is that a complex-valued function called the Riemann Zeta Function \( \zeta(s) \) quantifies the error in our approximations of \( \pi (x) \). Thus, knowing its behavior, particularly its zeros, tells us a lot about the distribution of the prime numbers. If we prove the Riemann hypothesis, we can perfectly predict the number of primes below any given magnitude, and, in doing so, achieve some semblance of mastery over the natural numbers. That's a big deal.

Longer Version

Here, we introduce the relevant technical context that elaborates the qualitative motivation given above. We assume an understanding of some basic complex analysis: Cauchy's and the Residue theorems, infinite and canonical products, analytic continuation of complex functions, and exposure to Fourier series. We begin with a formal statement of the main result, for \( z \in \mathbb{C} \) a nontrivial zero (ie. \(z \not\in \mathbb{C}\setminus\{ -2\mathbb{N}\} \) ). \[ \text{(Riemann Hypothesis.)} \ \ \zeta(z) = 0 \implies \text{Re}(z) = {1\over 2} \] More informally, it claims that all nontrivial zeros of the Riemann zeta function live on a vertical line of the complex plane defined by numbers with real part one half, where this line is often referred to as the "critical line."


[IN PROGRESS]

The Prime Number Theorem.

Infinite products link complex analysis to primes.

Zeta function basics.

Distributions.

Fourier series and the Riemann spectrum.

Linking the spectrum to Zeta zeros.

Conclusion.