On the Riemann Hypothesis, complex scalings and logarithmic time reversal

An approach to solve the Riemann Hypothesis is revisited within the framework of the special properties of Θ (theta) functions, and the notion of CT invariance. The conjugation operation C amounts to complex scaling transformations, and the T operation t→(1∕t) amounts to the reversal log(t)→−log(t). A judicious scaling-like operator is constructed whose spectrum Es=s(1−s) is real-valued, leading to s=12+iρ, and/or 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 zeros exist off the critical line. The role of the C,T transformations, and the properties of the Mellin transform of Θ functions were essential in our construction.