Prerequisites for this post: previous post, and complex analysis. For this entire post, $s$ is a complex variable with $s = \sigma + it$.

1. The $\Gamma$ function

So there’s this thing called the Gamma function. Denoted $\Gamma(s)$, it is defined by $$\Gamma(s) = \int_0^{\infty} x^{s-1} e^{-x} dx$$ as long as $\sigma > 0$. Here are its values at the first few integers:

$$\begin{aligned} \Gamma(1) &= 1 \\ \Gamma(2) &= 1 \\ \Gamma(3) &= 2 \\ \Gamma(4) &= 6 \\ \Gamma(5) &= 24. \end{aligned}$$

Prerequisites for this post: definition of Dirichlet convolution, and big $O$-notation.

Normally I don’t like to blog about something until I’m pretty confident that I have a reasonably good understanding of what’s happening, but I desperately need to sort out my thoughts, so here I go…

1. Primes

One day, an alien explorer lands on Earth in a 3rd grade classroom. He hears the teacher talk about these things called primes. So he goes up to the teacher and asks “how many primes are there less than $x$?”.

Answer: “uh. . .”.

Maybe that’s too hard, so the alien instead asks “about how many primes are there less than $x$?”.

This is again greeted with silence. Confused, the alien asks a bunch of the teachers, who all respond similarly, but then someone mentions that in the last couple hundred years, someone …

This is an expanded version of an answer I gave to a question that came up while I was assisting the 2014-2015 WOOT class. It struck me as an unusually good way to motivate higher math using stuff that people notice in high school but for some reason decide to not think about.

In high school precalculus, you’ll often be asked to find the roots of some polynomial with integer coefficients. For instance, $$x^3 - x^2 - x - 15 = (x-3)(x^2+2x+5)$$ has roots $3$, $1+2i$, $-1-2i$. Or as another example, $$x^3-3x^2-2x+2=(x+1)(x^2-4x+\mathrm{2\; \dots }$$