A Proof of the Riemann hypothesis The new version contains a change in Definition 3.4, resulting in simpler proofs of theorems in Sections 3 and 4. Also, important proof in Sec.5 has been modified, for completeness and clarity.

The function $G(z) = \int_0^\infty \xi^{z-1}(1+\exp(\xi))^{-1} \, d\xi$ is analytic and has the same zeros as the Riemann zeta function in the critical strip $D = \{z \in {\mathbf C} : 0 < \Re z < 1\}$. This paper combines some novel methods about indefinite integration, indefinite convolutions and inversions of Fourier transforms with numerical ranges of operators to prove the Riemann hypothesis.