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In this note we discuss explicitly the structure of two simple set of zeroswhich are associated with the mean staircase emerging from the zeta functionand we specify a solution using the Lambert W function. The argument of it maythen 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 afunction of an hermitean operator whose eigenvalues are the "trivial zeros" onthe 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 theusual trivial zeros $s = \rho + i \cdot t = - 2 n$, $n \geqslant 1$ integer)

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