标题： 格子和的零点：1. 批评线外的零点 显示英文标题

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

摘要： 考虑一类与Epstein zeta函数相关的二维和的零点，这类和的研究对象是Hejhal和Bombieri于1987年研究过的矩形格子上的和，特别是Potter和Titchmarsh于1935年首先研究过的某个和。 这些研究者证明了关于矩形格子上和的零点的若干性质，并指出某个特定和的零点偏离了临界线。 研究了矩形格子的周期比 $\lambda$ 的情况下一个此类零点的行为，结果表明它沿着一条轨迹连续演化，这条轨迹逐渐接近临界线，并在某一点达到临界线，而这一点是矩形格子和的二阶零点。 此外，还表明可以确定周期比 $\lambda$ 的某些范围，使得矩形格子和的零点偏离临界线。 显示英文摘要

摘要： 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.