On the Riemann Hypothesis and Complex Scalings

 An approach to solve the Riemann Hypothesis is revisited within the framework of the special properties of Θ $\Theta$ (theta) functions, and the notion of CT $\mathcal{C}\mathcal{T}$ invariance. The conjugation operation C $\mathcal{C}$ amounts to complex scaling transformations, and the T $\mathcal{T}$ operation t→(1∕t) $t\to (1∕t)$ amounts to the reversal log(t)→−log(t) $log\left(t\right)\to -log\left(t\right)$. A judicious scaling-like operator is constructed whose spectrum Es=s(1−s) $E_s=s(1-s)$ is real-valued, leading to s=12+iρ $s=\frac{1}{2}+i\rho$, and/or s=$s=$ real. These values are the location of the non-trivial and trivial zeta zeros, respectively. A thorough analysis of the one-to-one correspondence among the zeta zeros, and the orthogonality conditions among pairs of eigenfunctions, reveals that no $no$ zeros exist off the critical line. The role of the C,T $\mathcal{C},\mathcal{T}$ transformations, and the properties of the Mellin transform of Θ $\Theta$ functions were essential in our construction.