I know almost nothing on the subject, but I still feel that, to prove the prime number theorem for arithmetic progressions, the most delicate point remains, by far, the nonvanishing of $L(1,\chi)$. Would you agree with this formulation?

Dear Qiaochu Yuan: What is the OP? I think the way matrix exponential is usually presented, that is by invoking Jordan decomposition, is based on a conceptual error. This conceptual error (in my opinion) is the failure to see that the exponential of a matrix is obtained by evaluating on this matrix the product of the minimal polynomial f by the singular part of e^z/f(z). This statement is much simpler than the Jordan Decomposition Theorem.