Quasitransparent states in the logarithmic chain and nontrivial zeros of the Riemann zeta function

The Riemann hypothesis is a major open problem in number theory. It asserts that the nontrivial zeros of the Riemann zeta function (x) occur on the critical line, i. e. , for x=u+iy with u=12, in the complex plane. We discuss the physical realization of the zeros of (x) on the critical line by means of a 1D nonperiodic lattice with sites at positions z₉=dln (j+1) (j0, 1, 2,. . . ) along the z axis with d a length and the form factors of the sites modulated by (-1) ^j+1e^-{z₉} with playing the role of u. On the critical line, =12. We find quasitransparent states when y=kd2 is the imaginary part of a zero of (x) on the critical line with k the wave vector of the particle between sites z₉ and z₉+₁.