Albanian Journal of Mathematics (ISNN: 1930-1235), Vol 3, No 4 (2009)

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

Stefano Beltraminelli, Davide a Marca, Danilo Merlini

Abstract


In this note we discuss explicitly the structure of some simple sets of zeros which are associated with mean staircases emerging from the zeta function. They are given as solutions of an equation involving the Lambert W function. The argument of the latter function 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. In the general case, the sets of such trivial zeros are defined by the relation $S (t) \mathrel{\mathop:}= \frac{1}{\pi} \arg \left( \zeta \left( \frac{1}{2} + i t \right) \right)= \lambda$, $\lambda \in \mathbb{R} $ but we pay more attention to the case $\lambda = 0$ where $\tmop{Im} \left( \zeta \left( \frac{1}{2} + i
\cdot t \right) \right) = 0 \wedge \tmop{Re} \left( \zeta \left( \frac{1}{2} + i \cdot t \right) \right) \neq 0$ and to the case $\lambda = \frac{1}{2}$ given similarly exchanging the roles of Im and Re. (To distinguish from the usual trivial zeros $s = \rho + i \cdot t = - 2 n$, $n \geqslant 1$ integer)

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