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number theory and entropy

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K.H. Knuth, "Deriving laws from ordering relations", In: G.J. Erickson, Y. Zhai (eds.), Bayesian Inference and Maximum Entropy Methods in Science and Engineering, AIP Conference Proceedings 707 (2003) 204-235.

[author's description:] "In this paper I show that bi-valuations defined on distributive lattices give rise to a sum rule, a product rule and a Bayes' theorem analog that are most familiar in the realm of Bayesian probability theory. This work is a generalization of Richard T. Cox's derivation of probability theory from Boolean algebra by defining degrees of implication. However, here I show that the potential for application is much greater than previously envisioned. The Möbius function for the distributive lattice gives rise to Gian-Carlo Rota's inclusion-exclusion relation, which is responsible for the form of many laws familiar from areas of study as diverse as probability theory, number theory, geometry, information theory, and quantum mechanics."

K.H. Knuth, "Lattice duality: The origin of probability and entropy", Neurocomputing 67 C (2005) 245-274

[author's description:] "This paper shows how a straight-forward generalization of the zeta function of a distributive lattice gives rise to bi-valuations that represent degrees of belief in Boolean lattices of assertions and degrees of relevance in the distributive lattice of questions. The distributive lattice of questions originates from Richard T. Cox's definition of a question as the set of all possible answers, which I show is equivalent to the ordered set of down-sets of assertions. Thus the Boolean lattice of assertionns is shown to be dual to the distributive lattice of questions in the sense of Birkhoff's Representation Theorem. A straightforward correspondence between bi-valuations generalized from the zeta functions of each lattice give rise to bi-valuations that represent probabilities in the lattice of assertions and bi-valuations that represent entropies and higher-order informations in the lattice of questions."

P. Kumar, P.C. Ivanov, H.E. Stanley, "Information entropy and correlations in prime numbers"

[abstract:] "The difference between two consecutive prime numbers is called the distance between the primes. We study the statistical properties of the distances and their increments (the difference between two consecutive distances) for a sequence comprising the first 5 x 10⁷ prime numbers. We find that the histogram of the increments follows an exponential distribution with superposed periodic behavior of period three, similar to previously-reported period six oscillations for the distances."

Nature article on this research (24/03/03)

S.W. Golomb, "Probability, information theory, and prime number theory", Discrete Mathematics 106-107 (1992) 219-229

[abstract:] "For any probability distribution D = {\alpha(n)} on Z⁺, we define. . . the probability in D that a 'random' integer is a multiple of m; and . . . the probability in D that a 'random' integer is relatively prime to k. We specialize this general situation to three important families of distributions . . . Several basic results and concepts from analytic prime number theory are revisited from the perspective of these families of probability distributions, and the Shannon entropy for each of these families is determined."

C. Bonanno and M.S. Mega, "Toward a dynamical model for prime numbers" Chaos, Solitons and Fractals 20 (2004) 107-118

[abstract:] "We show one possible dynamical approach to the study of the distribution of prime numbers. Our approach is based on two complexity methods, the Computable Information Content and the Entropy Information Gain, looking for analogies between the prime numbers and intermittency."

The main idea here is that the Manneville map T_z exhibits a phase transition at z = 2, at which point the mean Algorithmic Information Content of the associated symbolic dynamics is n/log n. n is a kind of iteration number. For this to work, the domain of T_z [0,1] must be partitioned as [0,0.618...] U [0.618...,1] where 1.618... is the golden mean.

The authors attempt to exploit the resemblance to the approximating function in the Prime Number Theorem, and in some sense model the distribution of primes in dynamical terms, i.e. relate the prime number series (as a binary string) to the orbits of the Manneville map T₂. Certain refinements of this are then explored.

"We remark that this approach to study prime numbers is similar to the probabilistic approach introduced by Cramér...that is we assume that the [binary] string [generated by the sequence of primes]...is one of a family of strings on which there is a probability measure..."

E. Canessa, "Theory of analogous force on number sets" (preprint 07/03)

[abstract:] "A general statistical thermodynamic theory that considers given sequences of [natural numbers] to play the role of particles of known type in an isolated elastic system is proposed. By also considering some explicit discrete probability distributions p_x for natural numbers, we claim that they lead to a better understanding of probabilistic laws associated with number theory. Sequences of numbers are treated as the size measure of finite sets. By considering p_x to describe complex phenomena, the theory leads to derive a distinct analogous force f_x on number sets proportional to $(\fract{\partial p_{x}}{\partial x})_{T}$ at an analogous system temperature T. In particular, this yields to an understanding of the uneven distribution of integers of random sets in terms of analogous scale invariance and a screened inverse square force acting on the significant digits. The theory also allows to establish recursion relations to predict sequences of Fibonacci numbers and to give an answer to the interesting theoretical question of the appearance of the Benford's law in Fibonacci numbers. A possible relevance to prime numbers is also analyzed."

Informational theoretic entropy is defined in this setting in part II.B.

A.I. Aptekarev, J.S. Dehesa, A. Martinez-Finkelshtein, R. Yanez, "Discrete entropies of orthogonal polynomials" (preprint 10/2007)

[abstract:] "Let $p_n$ be the $n$-th orthonormal polynomial on the real line, whose zeros are $\lambda_j^{(n)}$, $j=1, ..., n$. Then for each $j=1, ..., n$, $$ \vec \Psi_j^2 = (\Psi_{1j}^2, ..., \Psi_{nj}^2) $$ with $$ \Psi_{ij}^2= p_{i-1}^2 (\lambda_j^{(n)}) (\sum_{k=0}^{n-1} p_k^2(\lambda_j^{(n)}))^{-1}, \quad i=1, >..., n, $$ defines a discrete probability distribution. The Shannon entropy of the sequence $\{p_n\}$ is consequently defined as $$ \mathcal S_{n,j} = -\sum_{i=1}^n \Psi_{ij}^{2} \log (\Psi_{ij}^{2}) . $$ In the case of Chebyshev polynomials of the first and second kinds an explicit and closed formula for $\mathcal S_{n,j}$ is obtained, revealing interesting connections with the number theory. Besides, several results of numerical computations exemplifying the behavior of $\mathcal S_{n,j}$ for other families are also presented."

P. Tempesta, "Group entropies, correlation laws and zeta functions" (preprint 05/2011)

[abstract:] "The notion of group entropy is proposed. It enables to unify and generalize many different definitions of entropy known in the literature, as those of Boltzmann–Gibbs, Tsallis, Abe and Kaniadakis. Other new entropic functionals are presented, related to nontrivial correlation laws characterizing universality classes of systems out of equilibrium, when the dynamics is weakly chaotic. The associated thermostatistics are discussed. The mathematical structure underlying our construction is that of formal group theory, which provides the general structure of the correlations among particles and dictates the associated entropic functionals. As an example of application, the role of group entropies in information theory is illustrated and generalizations of the Kullback–Leibler divergence are proposed. A new connection between statistical mechanics and zeta functions is established. In particular, Tsallis entropy is related to the classical Riemann zeta function."

[30] G. van der Geer and R. Schoof, "Effectivity of Arakelov divisors and the theta divisor of a number field", Selecta Math. (N.S.) 6 (2000) 377-398
[83] F.K. Schmidt, "Analytische Zahlentheorie in Körpern der Characteristik p", Math. Z. 33 (1931) 1-32
[96] A. Weil, "Sur l'analogie entre les corps de nombres algébraiques et les corps de functions algébreques", Revue Scient. 77 (1939) 104-106 (also in Collected Papers, Vol. I, Springer-Verlag, pp.236-240)

C. Chicchiero, Notes on symbolic dynamics, entropy, and prime numbers
 

I.J. Taneja, notes on entropy series, involving the Riemann zeta function

Prime Numbers and Entropy (Flash applet)

Irish teenager David Doherty has won the 2002 ESAT Young Scientist of the Year Award by proving a conjecture related to the distribution of primes. His continuing research project entitled "The Distribution of the Primes and the Underlying Order to Chaos" seeks to investigate parallels with thermodynamics, entropy, etc.