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- W3105294509 abstract "We study the {pair correlations between} prime numbers in an interval $M leq p leq M + L$ with $M rightarrow infty$, $L/M rightarrow beta > 0$. By analyzing the emph{structure factor}, we prove, conditionally on the {Hardy-Littlewood conjecture on prime pairs}, that the primes are characterized by unanticipated multiscale order. Specifically, their limiting structure factor is that of a union of an infinite number of periodic systems and is characterized by dense set of Dirac delta functions. Primes in dyadic intervals are the first examples of what we call {it effectively limit-periodic} point configurations. This behavior implies anomalously suppressed density fluctuations compared to uncorrelated (Poisson) systems at large length scales, which is now known as hyperuniformity. Using a scalar order metric $tau$ calculated from the structure factor, we identify a transition between the order exhibited when $L$ is comparable to $M$ and the uncorrelated behavior when $L$ is only logarithmic in $M$. Our analysis for the structure factor leads to an algorithm to reconstruct primes in a dyadic interval with high accuracy." @default.
- W3105294509 created "2020-11-23" @default.
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- W3105294509 date "2019-03-01" @default.
- W3105294509 modified "2023-09-30" @default.
- W3105294509 title "Hidden multiscale order in the primes" @default.
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- W3105294509 doi "https://doi.org/10.1088/1751-8121/ab0588" @default.
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