Notes on two lemmas concerning the Epstein zeta-function

Diananda, P. H.

doi:10.1017/s2040618500035036

Cited by 46 publications

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“…Right: the relative difference ζ2(I2, s) − ζ2(S hex , s) / |ζ2(S hex , s)|. It shows that the hexagonal lattice energy is lower than that of the square lattice for all s > 0, as it is proved in [198,48,79,65]. Figure 6.…”

Section: Resultsmentioning

confidence: 76%

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The crystallization conjecture: a review

Blanc¹,

Lewin²

2015

EMS Surv. Math. Sci.

142

191

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In this article we describe the crystallization conjecture. It states that, in appropriate physical conditions, interacting particles always place themselves into periodic configurations, breaking thereby the natural translation-invariance of the system. This famous problem is still largely open. Mathematically, it amounts to studying the minima of a real-valued function defined on R 3N where N is the number of particles, which tends to infinity. We review the existing literature and mention several related open problems, of which many have not been thoroughly studied.

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Section: Resultsmentioning

confidence: 76%

“…In dimension d = 2, it has been proved by Rankin [198], Cassels [48], Ennola [79] and Diananda [65], that the hexagonal lattice is the unique minimizer of the zeta function, for any s > 0, when the density is fixed. In other words, we have…”

Section: Optimal Lattices and Special Functionsmentioning

confidence: 99%

The crystallization conjecture: a review

Blanc¹,

Lewin²

2015

EMS Surv. Math. Sci.

142

191

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“…In dimension 2, W was shown to be minimized within the class of lattices of covolume 1 by the triangular lattice in [SS1], respectively [PS], by showing that this question could be reduced to the question of minimizing the Epstein zeta function, previously solved in [Cas,Ran,Enno1,Enno2,Dia,Mont] (see also [Ch, SaSt, OSP] for general dimension). In [SS1] it was conjectured, in view of the observations of triangular Abrikosov lattices of vortices in superconductors (and in line with the Cohn-Kumar conjecture) that the triangular lattice should minimize W in the 2D logarithmic case, among the class of all configurations of density 1.…”

Section: Coulomb and Riesz Interactions: Definitions And Motivationsmentioning

confidence: 99%

Crystallization for Coulomb and Riesz interactions as a consequence of the Cohn-Kumar conjecture

Petrache¹,

Serfaty

2020

Proc. Amer. Math. Soc.

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The Cohn-Kumar conjecture states that the triangular lattice in dimension 2, the E8 lattice in dimension 8, and the Leech lattice in dimension 24 are universally minimizing in the sense that they minimize the total pair interaction energy of infinite point configurations for all completely monotone functions of the squared distance. This conjecture was recently proved by Cohn-Kumar-Miller-Radchenko-Viazovska in dimensions 8 and 24. We explain in this note how the conjecture implies the minimality of the same lattices for the Coulomb and Riesz renormalized energies as well as jellium and periodic jellium energies, hence settling the question of their minimization in dimensions 8 and 24.Conjecture 1 (Cohn-Kumar [CK]). In dimension d = 2, 8, resp. 24, the lattice Λ 0 is universally minimizing in the sense that it minimizes E p among all possible point configurations of density 1 for all p's that are completely monotone functions of the squared distance.Coulangeon and Schürmann proved in [CS] a local version of this conjecture with the result that Λ 0 is a local minimizer of E p . Then Conjecture 1 was recently proved in dimensions 8 and 24 -it remains open in dimension 2.

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“…This question is in fact of number-theoretic nature: in [69] it is shown that the question in dimension 2 reduces to minimizing P p2ƒ jpj s , the Epstein zeta function, with s > 2 among lattices ƒ, a question that was in turn already solved in dimension 2 in the 1960s (see [17,22,26,27,60] and also [54] and references therein), but the same question is open in dimension d 3-except for d D 8 and d D 24-and it is only conjectured that the FCC is a local minimizer; see [71] and references therein. The connection with the minimization of W among lattices and that of the Epstein zeta functions is not even rigorously clear in dimension d 3.…”

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confidence: 99%

Higher‐Dimensional Coulomb Gases and Renormalized Energy Functionals

Rougerie

Serfaty

2015

Comm Pure Appl Math

183

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We consider a classical system of n charged particles in an external confining potential, in any dimension d ≥ 2. The particles interact via pairwise repulsive Coulomb forces and the coupling parameter is of order n −1 (mean-field scaling). By a suitable splitting of the Hamiltonian, we extract the next to leading order term in the ground state energy, beyond the mean-field limit. We show that this next order term, which characterizes the fluctuations of the system, is governed by a new "renormalized energy" functional providing a way to compute the total Coulomb energy of a jellium (i.e. an infinite set of point charges screened by a uniform neutralizing background), in any dimension. The renormalization that cuts out the infinite part of the energy is achieved by smearing out the point charges at a small scale, as in Onsager's lemma. We obtain consequences for the statistical mechanics of the Coulomb gas: next to leading order asymptotic expansion of the free energy or partition function, characterizations of the Gibbs measures, estimates on the local charge fluctuations and factorization estimates for reduced densities. This extends results of Sandier and Serfaty to dimension higher than two by an alternative approach.

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Notes on two lemmas concerning the Epstein zeta-function

Cited by 46 publications

References 4 publications

The crystallization conjecture: a review

The crystallization conjecture: a review

Crystallization for Coulomb and Riesz interactions as a consequence of the Cohn-Kumar conjecture

Higher‐Dimensional Coulomb Gases and Renormalized Energy Functionals

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