Generation of off-critical zeros for hypercubic Epstein zeta functions

Igor Travěnec and Ladislav Šamaj

Applied Mathematics and Computation, 2022, vol. 413, issue C

Abstract: We study the Epstein zeta-function formulated on the d-dimensional hypercubic lattice ζ(d)(s)=12∑′n1,…,nd(n12+…+nd2)−s/2 where the real part ℜ(s)>d and 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 d being a continuous variable ranging from 0 to ∞. Zeros of the Epstein zeta-function ρ=ρx+iρy are defined by ζ(d)(ρ)=0. The nontrivial zeros split into the “critical” zeros (on the critical line) with ρx=d2 and the “off-critical” zeros (off the critical line) with ρx≠d2. Numerical calculations reveal that the critical zeros form closed or semi-open curves ρy(d) which enclose disjunctive regions of the plane (ρx=d2,ρy). Each curve involves a number of left/right edge points ρ*=(d*2,ρy*), defined by a divergent tangent dρy/dd|ρ*. Every edge point gives rise to two conjugate tails of off-critical zeros with continuously varying dimension d which 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 d of the critical strip in the limit d→∞. As a by-product of the formalism, we derive an exact formula for limd→0ζ(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 2π/lnd in the limit d→∞.

Keywords: Epstein zeta-function; Hypercubic lattice; Analytic continuation; Jacobi theta function; Zeros; Critical phenomena (search for similar items in EconPapers)
Date: 2022
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Persistent link: https://EconPapers.repec.org/RePEc:eee:apmaco:v:413:y:2022:i:c:s0096300321006950

DOI: 10.1016/j.amc.2021.126611

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