Hamiltonians for the zeros of a general family of zeta functions

TL;DR

This paper constructs Hamiltonians whose eigenstates correspond to zeros of a broad class of zeta functions, advancing the Hilbert-Pólya approach to understanding these zeros.

Contribution

It introduces a general Hamiltonian framework that links eigenstates to zeros of various zeta functions, extending previous models to a wider family.

Findings

Hamiltonians constructed for zeros of general zeta functions

Eigenvalues relate to zeros via a specific transformation

Provides a new approach towards the Hilbert-Pólya conjecture

Abstract

Towards the Hilbert-P\'olya conjecture, in this paper, we present a general construction of Hamiltonian $\hat{H}_{f},$ which leads a general family of Hurwitz zeta functions $(-1)^{z_{n}-1}L(f,z_{n},x+1)$ defined by Mellin transform becomes their eigenstates under a suitable boundary condition, and the eigenvalues $E_{n}$ have the property that $z_{n}=\frac{1}{2}(1-iE_{n})$ are the zeros of a general family of zeta functions $L(f,z)\equiv L(f,z,1)$.