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.

Riemann Zeta on PostDoc Problems

Logarithms and primes