A proof of the Riemann hypothesis
  In this paper we study traces of an integral operator on two orthogonal subspaces of a $L^2$ space. Accidentally we discovered that one of the two traces is zero, which follows from the fact that the $S$-idele group $J_S$ is the disjoint union of $\xi I_S$'s with $\xi$ running through all $S$-units and $I_S$ being the fundamental domain for $J_S$. This discovery is also incidentally along the lines of Bombieri \cite[page 123, lines 12--14]{Bombieri2}. Another miracle in the argument is the simultaneous appearances of four integrals in the evaluation of traces. These four integrals all together made it possible to cancel all unwanted integration variables in evaluating traces of the operator on the two subspaces. We also used the disjoint decomposition of $J_S$ and the positivity of the convolution operator to prove that the trace of the integral operator on the second subspace is nonnegative. It follows that the integral operator has nonnegative trace on the $L^2$ space. This proves the positivity of Li's criterion \cite{Bombieri1}. By Li's criterion, all nontrivial zeros of the Riemann zeta-function lie on the critical line. 
Xian-Jin Li