2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/216940In 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 $\tmop{Im} (ζ({1/2} + i \cdot t)) = 0 \wedge \tmop{Re} (ζ({1/2} + i \cdot t)) \neq 0$ and viceversa for the second set. (To distinguish from the usual trivial zeros $s = ρ+ i \cdot t = - 2 n$, $n \geqslant 1$ integer)Number TheoryMathematical Physics11M26, 81Q10text