# Hamiltonians arising from L-functions in the Selberg class

@article{Suzuki2016HamiltoniansAF,
title={Hamiltonians arising from L-functions in the Selberg class},
author={Masatoshi Suzuki},
journal={arXiv: Number Theory},
year={2016}
}
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 matrix valued function called a Hamiltonian. To establish the equivalent condition, we solve and use an inverse spectral problem for canonical systems of special type.
2 Citations
An inverse problem for a class of diagonal Hamiltonians
Hamiltonians are 2-by-2 positive semidefinite real symmetric matrix valued functions satisfying certain conditions. In this paper, we solve the inverse problem for which recovers a Hamiltonian fromExpand
Integral operators arising from the Riemann zeta function
In this paper we have two issues coming from the same background. The first one is to describe a certain ratio of Fredholm determinants of integral operators arising from the Riemann zeta function byExpand

#### References

SHOWING 1-10 OF 50 REFERENCES
L-FUNCTIONS AS DISTRIBUTIONS
• 2015
We define an axiomatic class of L-functions extending the Selberg class. We show in particular that one can recast the traditional conditions of an Euler product, analytic continuation and functionalExpand
On the structure of the Selberg class, VII: 1
• Mathematics
• 2011
The Selberg class S is a rather general class of Dirichlet series with functional equation and Euler product and can be regarded as an axiomatic model for the global L-functions arising from numberExpand
An inverse problem for a class of canonical systems and its applications to self-reciprocal polynomials
A canonical system is a kind of first-order system of ordinary differential equations on an interval of the real line parametrized by complex numbers. It is known that any solution of a canonicalExpand
A spectral interpretation for the zeros of the Riemann zeta function
Based on work of Alain Connes, I have constructed a spectral interpretation for zeros of L-functions. Here we specialise this construction to the Riemann zeta function. We construct an operator on aExpand
Trace formula in noncommutative geometry and the zeros of the Riemann zeta function
Abstract. We give a spectral interpretation of the critical zeros of the Riemann zeta function as an absorption spectrum, while eventual noncritical zeros appear as resonances. We give a geometricExpand
On the nature of the de Branges Hamiltonian
We prove the theorem announced by the author in 1995 in the paper “A criterion for the discreteness of the spectrum of a singular canonical system” (Funkts. Anal. Prilozhen., 29, No. 3).In developingExpand
Pontryagin spaces of entire functions I
• Mathematics
• 1999
We give a generalization of L.de Branges theory of Hilbert spaces of entire functions to the Pontryagin space setting. The aim of this-first-part is to provide some basic results and to investigateExpand
Value-Distribution of L-Functions
Dirichlet Series and Polynomial Euler Products.- Interlude: Results from Probability Theory.- Limit Theorems.- Universality.- The Selberg Class.- Value-Distribution in the Complex Plane.- The RiemannExpand
On the prime number theorem for the Selberg class
• 2003
Abstract. We show that the prime number theorem is equivalent with the non-vanishing on the 1-line, in the general setting of the Selberg class $\mathcal{S}$ of $\mathcal{L}$-functions. The proofExpand
On a representation of the idele class group related to primes and zeros of L-functions
Let K be a global field. Using natural spaces of functions on the adele ring and the idele class group of K, we construct a virtual representation of the idele class group of K whose character isExpand