Zeros of Lattice Sums: 1. Zeros off the Critical Line

Zeros of two-dimensional sums of the Epstein zeta type over rectangular lattices of the type investigated by Hejhal and Bombieri in 1987 are considered, and in particular a sum first studied by Potter and Titchmarsh in 1935. These latter proved several properties of the zeros of sums over the rectangular lattice, and commented on the fact that a particular sum had zeros off the critical line. The behaviour of one such zero is investigated as a function of the ratio of the periods $\lambda$ of the rectangular lattice, and it is shown that it evolves continuously along a trajectory which approaches the critical line, reaching it at a point which is a second-order zero of the rectangular lattice sum. It is further shown that ranges of the period ratio $\lambda$ can be so identified for which zeros of the rectangular lattice sum lie off the critical line.