Hidden multiscale order in the primes

Torquato, Salvatore; Zhang, G.; Courcy-Ireland, Matthew de

doi:10.1088/1751-8121/ab0588

Cited by 34 publications

(44 citation statements)

References 63 publications

(206 reference statements)

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“…Hence, the primes become effectively limit-periodic, despite the variable pattern of occupied sites, as proved in Ref. [27] and illustrated in Fig. 3.…”

Section: Resultsmentioning

confidence: 56%

“…The proof of (3) is given in Ref. [27]. It is based on grouping the terms in the sum defining S(πm/n) according to their remainder after division by 2n.…”

Section: Resultsmentioning

confidence: 99%

“…We prove that the primes are characterized by unanticipated multiscale order; see Ref. [27] for details. Specifically, an analytical formula that we derive for their limiting structure factor S(k) has dense Bragg peaks, as in the case of quasicrystals [2].…”

Section: Introductionmentioning

confidence: 84%

“…An important aspect of the distribution of prime numbers is that larger ones become increasingly sparse. The drop-off is gradual enough that, in Gallagher's regime of short intervals or even for the longer intervals considered here, the density of prime numbers can be treated as constant [27]. According to the prime number theorem [28], the prime counting function π(x), which gives the number of primes less than x, in the large-x asymptotic limit is given by…”

Section: Introductionmentioning

confidence: 97%

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Uncovering multiscale order in the prime numbers via scattering

2018

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The prime numbers have been a source of fascination for millenia and continue to surprise us. Motivated by the hyperuniformity concept, which has attracted recent attention in physics and materials science, we show that the prime numbers in certain large intervals possess unanticipated order across length scales and represent the first example of a new class of many-particle systems with pure point diffraction patterns, which we call effectively limit-periodic. In particular, the primes in this regime are hyperuniform. This is shown analytically using the structure factor S(k), proportional to the scattering intensity from a many-particle system. Remarkably, the structure factor for primes is characterized by dense Bragg peaks, like a quasicrystal, but positioned at certain rational wavenumbers, like a limit-periodic point pattern. We identify a transition between ordered and disordered prime regimes that depends on the intervals studied. Our analysis leads to an algorithm that enables one to predict primes with high accuracy. Effective limit-periodicity deserves future investigation in physics, independent of its link to the primes.

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“…Hence, the primes become effectively limit-periodic, despite the variable pattern of occupied sites, as proved in Ref. [27] and illustrated in Fig. 3.…”

Section: Resultsmentioning

confidence: 56%

“…The proof of (3) is given in Ref. [27]. It is based on grouping the terms in the sum defining S(πm/n) according to their remainder after division by 2n.…”

Section: Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 84%

Section: Introductionmentioning

confidence: 97%

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Uncovering multiscale order in the prime numbers via scattering

2018

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“…† correspondence sent to: xwxfat@gmail.com ‡ correspondence sent to: mohanchen@pku.edu.cn § correspondence sent to: yang.jiao.2@asu.edu ¶ correspondence sent to: hzhuang7@asu.edu mune systems [27], amorphous silicon [28,29], a wide class of disordered cellular materials [30], dynamic random organizating systems [31][32][33][34][35], and even the distribution of primes on the number axis [36]. In addition, it has been shown that hyperuniform materials can be designed to possess superior physical properties including large isotropic photonic band gaps [37][38][39], optimized transport properties [40], mechanical properties [41] as well as optimal multi-functionalities [42].…”

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confidence: 99%

Disordered hyperuniformity in two-dimensional amorphous silica

et al. 2020

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Disordered hyperuniformity (DHU) is a recently proposed new state of matter, which has been observed in a variety of classical and quantum many-body systems. DHU systems are characterized by vanishing infinite-wavelength density fluctuations and are endowed with unique novel physical properties. Here we report the first discovery of disordered hyperuniformity in atomic-scale 2D materials, i.e., amorphous silica composed of a single layer of atoms, based on spectral-density analysis of high-resolution transmission electron microscope images. Subsequent simulations suggest that the observed DHU is closely related to the strong topological and geometrical constraints induced by the local chemical order in the system. Moreover, we show via large-scale density functional theory calculations that DHU leads to almost complete closure of the electronic band gap compared to the crystalline counterpart, making the material effectively a metal. This is in contrast to the conventional wisdom that disorder generally diminishes electronic transport and is due to the unique electron wave localization induced by the topological defects in the DHU state.Disorder hyperuniform (DHU) systems are a unique class of disordered systems which suppress large-scale density fluctuations like crystals and yet possess no Bragg peaks [1, 2]. For a point configuration (e.g., a collection of particle centers of a many-body system), hyperuniformity is manifested as the vanishing structure factor in the infinite-wavelength (or zero-wavenumber) limit, i.e., lim k→0 S(k) = 0, where k = 2π/λ is the wavenumber. In this case of a random field, the hyperuniform condition is given by lim k→0ψ (k) = 0, whereψ(k) is the spectral density [2]. It has been suggested that hyperuniformity can be considered as a new state of matter [1], which possesses a hidden order in between of that of a perfect crystal and a totally disordered system (e.g. a Poisson distribution of points).Recently, a wide spectrum of physical and biological systems have been identified to possess the remarkable property of hyperuniformity, which include the density fluctuations in early universe [3], disordered jammed packing of hard particles [4][5][6][7], certain exotic classical ground states of many-particle systems [8][9][10][11][12][13][14][15], jammed colloidal systems [16][17][18][19], driven non-equilibrium systems [20][21][22][23], certain quantum ground states [24,25], avian photoreceptor patterns [26], organization of adapted im- * These authors contributed equally to this work. † correspondence sent to: xwxfat@gmail.com ‡ correspondence sent to: mohanchen@pku.edu.cn § correspondence sent to: yang.jiao.2@asu.edu ¶ correspondence sent to: hzhuang7@asu.edu mune systems [27], amorphous silicon [28, 29], a wide class of disordered cellular materials [30], dynamic random organizating systems [31-35], and even the distribution of primes on the number axis [36]. In addition, it has been shown that hyperuniform materials can be designed to possess superior physical properties including ...

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Predicting transport characteristics of hyperuniform porous media via rigorous microstructure-property relations

Torquato

2020

Advances in Water Resources

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Hidden multiscale order in the primes

Cited by 34 publications

References 63 publications

Uncovering multiscale order in the prime numbers via scattering

Uncovering multiscale order in the prime numbers via scattering

Disordered hyperuniformity in two-dimensional amorphous silica

Predicting transport characteristics of hyperuniform porous media via rigorous microstructure-property relations

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