Hamiltonians arising from L-functions in the Selberg class

TL;DR

This paper links the Grand Riemann Hypothesis for certain L-functions to canonical systems of differential equations, providing a new equivalent condition through inverse problems involving Hamiltonians.

Contribution

It introduces a novel equivalence between the Grand Riemann Hypothesis and properties of canonical systems derived from L-functions in the Selberg class.

Findings

Established a new equivalent condition for the Grand Riemann Hypothesis.

Connected L-functions in the Selberg class to canonical systems via inverse problems.

Provided a framework to analyze L-functions through differential equations.

Abstract

We establish a new equivalent condition for the Grand Riemann Hypothesis for L-functions in a wide subclass of the Selberg class in terms of canonical systems of differential equations. A canonical system is determined by a real symmetric matrixvalued function called a Hamiltonian. To establish the equivalent condition, we use an inverse problem for canonical systems of a special type.