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

In this note we discuss explicitly the structure of some simple setsof values on the critical line (the rivial critical values") which are associatedwith the mean staircases emerging from the Zeta function. They are givenas solutions of an equation involving the Lambert W-function. The argumentof the latter function may then be set equal to a special N 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 eigenvaluesare exactly these values. In the general case, the sets of such trivial criticalvalues (zeros) are special solutions of a parametric equation involving a linearcombination of Re (s) and Im (s) on the critical line.