Nearest-neighbor-spacing distribution of prime numbers and quantum chaos

Wolf, Marek

doi:10.1103/physreve.89.022922

Cited by 24 publications

(33 citation statements)

References 44 publications

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“…On the other hand, the gaps between zeroes present a remarkably well defined trend, displaying an exponential decay according to the size, the length of the gap and the frequency with which it appears. This important result echoes previous studies regarding gaps between primes, such as those presented in [33,[36][37][38][39][40][41][42]. Whether or not this will be true for any given interval is another open question.…”

Section: Discussionsupporting

confidence: 85%

“…Recent computational work points that "after appropriate rescaling, the statistics of spacings between adjacent prime numbers follows the Poisson distribution" [41]. See [41,42] and references therein for more on the statistics of the gaps between consecutive prime numbers. The question at hand is encapsulated by whether after a given number all turns in the Ladder will have an "order".…”

Section: Discussionmentioning

confidence: 99%

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Jacob’s Ladder: Prime Numbers in 2D

Fraile

Martínez

Fernández

2020

MCA

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Prime numbers are one of the most intriguing figures in mathematics. Despite centuries of research, many questions remain still unsolved. In recent years, computer simulations are playing a fundamental role in the study of an immense variety of problems. In this work, we employ that to present a simple representation of prime numbers in two dimensions that allows us to formulate a number of conjectures that may lead to important avenues in the field of research on prime numbers. In particular, although the zeroes in our representation grow in a somewhat erratic, hardly predictable way, the gaps between them present a remarkable property: there is a clear exponential decay in the frequency of the gaps vs gap size. The smaller the gaps, the more frequently they appear.

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Section: Discussionsupporting

confidence: 85%

Section: Discussionmentioning

confidence: 99%

Jacob’s Ladder: Prime Numbers in 2D

Fraile

Martínez

Fernández

2020

MCA

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“…Other indicators can be used to define dynamical systems. The Lyapunov exponent (Wolf et al 1985;Wolf 2014) indicates chaos when large and positive, but this application is difficult, even impossible for experimental data records. (Sprott 2003).…”

Section: The Temporal Fractality Of Precipitationmentioning

confidence: 99%

Definition of a temporal distribution index for high temporal resolution precipitation data over Peninsular Spain and the Balearic Islands: the fractal dimension; and its synoptic implications

et al. 2018

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Precipitation on the Spanish mainland and in the Balearic archipelago exhibits a high degree of spatial and temporal variability, regardless of the temporal resolution of the data considered. The fractal dimension indicates the property of self-similarity, and in the case of this study, wherein it is applied to the temporal behaviour of rainfall at a fine (10-minute) resolution from a total of 48 observatories, it provides insights into its more or less convective nature. The methodology of Jenkinson and Collison which automatically classifies synoptic situations at the surface, as well as an adaptation of this methodology at 500 hPa, was applied in order to gain insights into the synoptic implications of extreme values of the fractal dimension. The highest fractal dimension values in the study area were observed in places with precipitation that has a more random behaviour over time with generally high totals. Four different regions in which the atmospheric mechanisms giving rise to precipitation at the surface differ from the corresponding above-ground mechanisms have been identified in the study area based on the fractal dimension. In the north of the Iberian Peninsula, high fractal dimension values are linked to a lower frequency of anticyclonic situations, whereas the opposite occurs in the central region. In the Mediterranean, higher fractal dimension values are associated with a higher frequency of the anticyclonic type and a lower frequency of the advective type from the east. In the south, lower fractal dimension values occur with more frequent anticyclonic-easterly hybrid weather types and with less frequent cyclonic type.

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“…Specifically, for maximal gaps G c between primes p = r + nq ∈ P c in k-tuples (k ≥ 2), the gap sizes are mostly found in the neighborhood of the corresponding trend curves of Eqs. (12), (14) derived from extreme value theory. However, for k = 1, the trend Eq.…”

Section: The Growth Trend Of Maximal Gapsmentioning

confidence: 99%

“…Maximal gaps G c between prime sextuplets p = r + nq ∈ P c below x for q = 313, x < 10 14 , k = 6. Dotted curves: trends T c (•) andT c ( ) of Equations (12) and(14); top line: y = C −1 6 ϕ 6 (q) log 7 p.…”

mentioning

confidence: 99%