Abstract
We show how the cross-disciplinary transfer of techniques from dynamical systems theory to number theory can be a fruitful avenue for research. We illustrate this idea by exploring from a nonlinear and symbolic dynamics viewpoint certain patterns emerging in some residue sequences generated from the prime number sequence. We show that the sequence formed by the residues of the primes modulo k are maximally chaotic and, while lacking forbidden patterns, unexpectedly display a non-trivial spectrum of Renyi entropies which suggest that every block of size , while admissible, occurs with different probability. This non-uniform distribution of blocks for contrasts Dirichlet’s theorem that guarantees equiprobability for . We then explore in a similar fashion the sequence of prime gap residues. We numerically find that this sequence is again chaotic (positivity of Kolmogorov–Sinai entropy), however chaos is weaker as forbidden patterns emerge for every block of size . We relate the onset of these forbidden patterns with the divisibility properties of integers, and estimate the densities of gap block residues via Hardy–Littlewood k-tuple conjecture. We use this estimation to argue that the amount of admissible blocks is non-uniformly distributed, what supports the fact that the spectrum of Renyi entropies is again non-trivial in this case. We complete our analysis by applying the chaos game to these symbolic sequences, and comparing the Iterated Function System (IFS) attractors found for the experimental sequences with appropriate null models.
Highlights
Number theory is a millennial branch of pure mathematics devoted to the study of the integers, whose implications and tentacles—despite the expectations of theorists like Dickson or Hardy— pervade today almost all areas of mathematics but are at the basis of many technological applications
Before exploring from a dynamical systems viewpoint whether the set of sequences defined above evidence some degree of order, we should have an idea of what kind of patterns could naturally emerge in such sequences, even if no actual structure was present
From a dynamical systems viewpoint, these results suggest that the gap residue sequence manifests chaotic behaviour, but of weaker intensity as h(0) < log p
Summary
Number theory is a millennial branch of pure mathematics devoted to the study of the integers, whose implications and tentacles—despite the expectations of theorists like Dickson or Hardy— pervade today almost all areas of mathematics but are at the basis of many technological applications. Starting from the celebrated coincidence between Montgomery’s work on the statistics of the spacings between Riemann zeta zeros and Dyson’s analogous work on eigenvalues of random matrices in nuclear physics [6,7,8], we have seen how number theory and physics have built bridges between each other These connections range from the reinterpretation of the Riemann zeta function as a partition function [9] or the focus of the Riemann hypothesis via quantum chaos [10], to multifractality in the distribution of primes [11], computational phase transitions and criticality originating in combinatoric problems [12,13,14,15,16], or stochastic network models of primes and composites [17] to cite only a few examples (see [5] for an extensive bibliography)
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