On the Riemann Hypothesis, Complex Scalings and Logarithmic Time Reversal

November 22, 2017
3:00pm to 4:00pm

IFT Seminar Room/Red Room

Specialist level
Carlos Castro Perelman
Clark Atlanta U.

IFT Seminar Room/Red Room


An approach to solving 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 = 1 +iρ, and/or 2 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.