Riemann Hypothesis: Architecture of a conjecture "along" the lines of Pólya. From trivial zeros and Harmonic Oscillator to information about non-trivial zeros of the Riemann zeta-function

Abstract

We propose an architecture of a conjecture concerning the Riemann Hypothesis in the form of an "alternative" to the Pólya strategy: we construct a Hamiltonian H_Polya whose spectrum coincides exactly with that of the Harmonic Oscillator Hamiltonian H_osc if and only if the Riemann Hypothesis holds true. In other words, it can be said that we formulate the Riemann Hypothesis by means of a non-commutative structure on the real axis, viz., that of the Harmonic Oscillator, by an equation of the type H_Polya(H_osc) = H_osc: the Harmonic Oscillator operator, if viewed as an argument of H_Polya, reproduces itself.

7 pages

References in corpus (3)

• Landau levels and Riemann zeros
• A few equalities involving integrals of the logarithm of the Riemann zeta-function and equivalent to the Riemann hypothesis II
• A few equalities involving integrals of the logarithm of the Riemann zeta-function and equivalent to the Riemann hypothesis III. Exponential weight functions