On the Riemann Hypothesis, complex scalings and logarithmic time reversal

Abstract

Abstract An approach to solve the Riemann Hypothesis is revisited within the framework of the special properties of Θ (theta) functions, and the notion of C T invariance. The conjugation operation C amounts to complex scaling transformations, and the T operation t → ( 1 ∕ t ) amounts to the reversal l o g ( t ) → − l o g ( t ) . A judicious scaling-like operator is constructed whose spectrum E s = s ( 1 − s ) is real-valued, leading to s = 1 2 + i ρ , 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 n o 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.