Intros
In this series, I plan to upload very short introductions to beautiful mathematical ideas and subjects. The goal is to skip over as much detail as possible, while providing a range of motivational examples and outlining the underlying ideas.-
Dirichlet series: They can be used to answer questions such as the following: given a (number theoretic) sequence \(a_1,a_2,\dots\) of nonnegative integers, what is the asymptotic behavior of \(\sum_{n\leq N}a_n\) as \(N\) goes to infinity? We in particular cover the following sequences: $$ \begin{aligned} \sigma_n &= \text{number of divisors of $n$},\\ \Delta_n &= \#\{(k,f_1,\dots,f_k)\mid k\geq0,\quad f_1,\dots,f_k\geq2,\quad n=f_1\cdots f_k\}\\ &= \text{number of ordered factorizations of $n$ into any number of factors bigger than 1},\\ \mathbb P_n &= \text{1 if $n$ is prime, 0 otherwise}.\qquad\text{(We sketch how to prove the Prime Number Theorem.)} \end{aligned} $$ Prerequisites: Basic Number Theory, Complex Analysis, (familiarity with Ordinary Generating Functions a plus)
Software
- Sauklaue, an application for hand-written online lecturing using an external graphics tablet.
- Contributed to CMS, a contest management system for informatics olympiads, as well as to the fork used at German informatics olympiad training camps.
- Chess clock with an undo button