Mean Staircase of the Riemann Zeros: a comment on the Lambert W function and an algebraic aspect

TL;DR

This paper explores the structure of specific zeros related to the Riemann zeta function's mean staircase, utilizing the Lambert W function and matrix models to connect zeros with eigenvalues of Hermitian operators.

Contribution

It introduces a novel algebraic framework linking zeros of the zeta function to the Lambert W function and matrix models, providing explicit solutions and new insights into their structure.

Findings

Explicit solutions for zeros using Lambert W function

Connection between zeros and eigenvalues of Hermitian operators

Representation of zeros via classical matrix models

Abstract

In this note we discuss explicitly the structure of two simple set of zeros which are associated with the mean staircase emerging from the zeta function and we specify a solution using the Lambert W function. The argument of it may then be set equal to a special $N \times N$ classical matrix (for every $N$) related to the Hamiltonian of the Mehta-Dyson model. In this way we specify a function of an hermitean operator whose eigenvalues are the "trivial zeros" on the critical line. The first set of trivial zeros is defined by the relations $\tmop{Im} (\zeta ({1/2} + i \cdot t)) = 0 \wedge \tmop{Re} (\zeta ({1/2} + i \cdot t)) \neq 0$ and viceversa for the second set. (To distinguish from the usual trivial zeros $s = \rho + i \cdot t = - 2 n$, $n \geqslant 1$ integer)