Iterative map with power-law scaling of Gamma-distributed fluctuations related to prime numbers
Abstract
Prime numbers arise in several contexts beyond number theory, including statistical mechanics, quantum mechanics, and dynamical systems. However, the mechanisms underlying the irregularities of their sequence and their connections to physical systems remain poorly understood. The present work provides further insight into the search for deterministic fingerprints in the prime sequence. To this end, prime gaps at different separation distances are investigated through an empirical analysis. Based on this analysis, a modified local approximation of the Prime Number Theorem is introduced and analyzed empirically, in which the logarithmic gap estimate is evaluated at a midpoint-corrected argument. From this relation an iterative map is obtained that reproduces the classical asymptotic prime spacing at leading order and satisfies an approximate semigroup property. The residual fluctuations are found to follow Gamma distributions rescaled by the correction terms, with a variance exhibiting an approximate power-law scaling in the prime separation distance.
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BibTeX
@misc{ciszak2026iterative,
title = {{Iterative map with power-law scaling of Gamma-distributed fluctuations related to prime numbers}},
author = {Marzena Ciszak},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2605.17622},
eprint = {2605.17622},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2605.17622},
}
RIS
TY - UNPB
TI - Iterative map with power-law scaling of Gamma-distributed fluctuations related to prime numbers
AU - Marzena Ciszak
PY - 2026
DA - 2026/09/28
PB - arXiv
DO - 10.48550/arxiv.2605.17622
UR - https://arxiv.org/abs/2605.17622
AB - Prime numbers arise in several contexts beyond number theory, including statistical mechanics, quantum mechanics, and dynamical systems. However, the mechanisms underlying the irregularities of their sequence and their connections to physical systems remain poorly understood. The present work provides further insight into the search for deterministic fingerprints in the prime sequence. To this end, prime gaps at different separation distances are investigated through an empirical analysis. Based on this analysis, a modified local approximation of the Prime Number Theorem is introduced and analyzed empirically, in which the logarithmic gap estimate is evaluated at a midpoint-corrected argument. From this relation an iterative map is obtained that reproduces the classical asymptotic prime spacing at leading order and satisfies an approximate semigroup property. The residual fluctuations are found to follow Gamma distributions rescaled by the correction terms, with a variance exhibiting an approximate power-law scaling in the prime separation distance.
ER -