Spaces of Dirichlet Series
Note: this was originally a thread on my twitter Let's talk about a weird connection between functional analysis and number theory: Dirichlet Series! (Full Disclosure: I'm not a number theorist, not even an analytic number theorist, but spaces of functions are my jam.) The most famous Dirichlet series is the Riemann zeta function, but we can generalize by changing the coefficients: To help get our hands on these, let's consider a very simple case where we don't have to be distressed about convergence: \(2^{-s}\). Looking at how this maps the plane helps me think about things like periodicity (which gets complicated when looking at a full DS). Complex TAYLOR series converge in disks in \(\mathbb{C}\). Dirichlet series converge in right half planes of \(\mathbb{C}\). Show this by showing that if a DS converges at a point \(s_0\), it converges in a sector, then union those to get a half plane. So instead of a radius of convergence like we get for Taylor se