Number Theory on PostDoc Problems

Logarithms and primes

Thu, 26 Oct 2023 09:01:26 +0000

I came across this video by NumberPhile and 3Blue1Brown about the density of primes that are 1 mod 4 vs the density of primes that are 3 mod 4. The main fact is quite cool but there was something that he said during in the middle of the video that I found more interesting.

Define $\chi : \mathbb{N} \to {-1, 0, 1}$ as $$ \chi(n) = \begin{cases} 1 & \text{ if } n \equiv 1 \mod 4 \ -1 & \text{ if } n \equiv 3 \mod 4 \ 0 & \text{ otherwise. } \end{cases} $$ The first interesting fact that I do not know how to prove: $$ \sum \limits_{n \in \mathbb{N}} \dfrac{\chi(n)}{n} = \dfrac{\pi}{4}. $$ Note that the series is only conditionally convergent so the order of summation matters.

The L-functions and modular forms database (LMFDB)

Sun, 24 Sep 2023 12:22:43 +0000

LMFDB Github

The LMFDB is a database of mathematical objects arising in number theory and arithmetic geometry that illustrates some of the mathematical connections predicted by the Langlands program.

I love this website. I don’t understand anything on it. But I’m amazed that this exists and I marvel at the amount of effort the writers must have put into making this happen. I’ve always thought that number theory is unique in how it manages to straddle the concrete and the abstract worlds so effortlessly. This website is a proof of this.

Every Prime $\equiv 1 \mod 4$ Is a Sum of Two Squares

Sun, 10 Sep 2023 15:35:44 +0000

I came across this “one sentence” proof that every prime congruent to 1 mod 4 can be written as a sum of two squares:

https://doi.org/10.2307/2323918

The proof involves constructing the following cryptic involution $$ (x, y, z) \mapsto \begin{cases}(x+2 z, z, y-x-z) & \text { if } x<y-z \ (2 y-x, y, x-y+z) & \text { if } y-z<x<2 y \ (x-2 y, x-y+z, y) & \text { if } x>2 y\end{cases} $$ on the set $S=\left{(x, y, z) \in \mathbb{N}^{3}: x^{2}+4 y z=p\right}$ and showing that the involution has exactly one fixed point $(1, 1, k)$ if and only if $p$ is a prime of the form $4k + 1$.