Metamaterials and Cesàro convergence

In this paper, we show that the linear dielectrics and magnetic materials in matter obey a special kind of mathematical property known as Cesàro convergence. Then, we also show the analytical continuation of permittivity and permeability to the complex plane in terms of the Riemann zeta (ζ) function. The nontrivial zeros on the half-line of the Riemann zeta (ζ) function correspond to permittivity ξe = 0 and permeability ξm = 0. The permittivity ξe = 0 and permeability ξm = 0 in the literature are known as zero index materials.