Generation of off-critical zeros for hypercubic Epstein zeta functions

We study the Epstein zeta-function formulated on the d-dimensional hypercubic lattice zeta((d))(s) = 1/2 Sigma' (n1,...,nd) (n(1)(2) +... + n(d)(2))(-s/2) where the real part R(s) > dand the summation runs over all integers except of the origin (0, 0,..., 0). An analytical continuation of the Epstein zeta-function to the whole complex s-plane is constructed for the spatial dimension dbeing a continuous variable ranging from 0 to infinity. Zeros of the Epstein zeta-function rho = rho x + i rho(y) are defined by zeta((d)) (rho) = 0. The nontrivial zeros split into the critical zeros (on the critical line) with rho(x) = d/2 and the off-critical zeros (offthe critical line) with rho(x) not equal d/2. Numerical calculations reveal that the critical zeros form closed or semi-open curves rho(y) (d) which enclose disjunctive regions of the plane (rho(x) = d/2,rho(y)). Each curve involves a number of left/right edge points rho*= (d */2,rho*(y)), defined by a divergent tangent d(rho y)/dd|(rho'). Every edge point gives rise to two conjugate tails of off-critical zeros with continuously varying dimension dwhich exhibit a singular expansion around the edge point, in analogy with critical phenomena for second-order phase transitions. For each dimension d > 9.24555... there exists a conjugate pair of real off-critical zeros which tend to the boundaries 0 and dof the critical strip in the limit d ->infinity. As a by-product of the formalism, we derive an exact formula for lim(d -> 0)zeta((d)) (s)/d. An equidistant distribution of critical zeros along the imaginary axis is obtained for large d, with spacing between the nearest-neighbour zeros vanishing as 2 pi/ ln din the limit d ->infinity. (C) 2021 Elsevier Inc. All rights reserved.

Automated summary

This study investigates the Epstein zeta-function on a d-dimensional hypercubic lattice and analyzes the distribution of critical and off-critical zeros in different dimensions. Numerical calculations reveal patterns of zero distribution and behavior around critical points in the complex s-plane.