Euler–Riemann–Dirichlet Lattices: Applications of η ( s ) Function in Physics

We discuss applications of the Dirichlet η ( s ) function in physics. To this end, we provide an introductory description of one-dimensional (1D) ionic crystals, which are well-known in the condensed matter physics literature, to illustrate the central issue of the paper: A generalization of the Coulomb interaction between alternating charges in such crystalline structures. The physical meaning of the proposed form, characterized by complex (in the mathematical sense) ion–ion interactions, is argued to have emerged in many-body systems, which may include effects from vacuum energy fluctuations. We first consider modifications to the bare Coulomb interaction by adding an imaginary component to the exponent of the Coulomb law of the form s = 1 + i b , where b is a real number. We then extend the results to slower-decaying interactions, where the exponent becomes s = a + i b , presenting numerical results for values 1 / 2 ≤ a ≤ 2 , which include the critical strip relevant to the Riemann hypothesis scenario.