Number Theory Seminar

The classical works of Euler, Dirichlet, Hermite, Lindemann, Gelfond, and Baker established the transcendence of $L(k,\chi)$ for $k=1$ and all nonprincipal Dirichlet characters $\chi$, as well as for the pairs $L(k,\chi)$ of a positive integer $k$ and a Dirichlet character $\chi$ of the same parity: $\chi(-1) = (-1)^k$. To this day, only one further special value has been proven irrational: Apery's 1978 miraculous rational approximations to $\zeta(3)$ converged fast enough to provide such a proof. In this talk, I will explain a new result joint with Frank Calegari and Yunqing Tang, which in particular adds a further special value to the irrationality proofs list.