probability, statistics, random walks, and Brownian motion
in connection with number theory

thought-proving quotes

historical background - H. Cramér's contribution to the field

(approximately) chronological bibliography

papers by Khrennikov, et al. on p-adic probability, etc.

number theoretic applications of random matrix theory

thought-provoking quotes

"It is evident that the primes are randomly distributed but, unfortunately, we don't know what 'random' means."

R.C. Vaughan (February 1990)

"Consider the integers divisible by both p and q [p and q both prime]. To be divisible by p and q is equivalent to being divisible by pq and consequently the density of the new set is 1/pq. Now, 1/pq = 1/p * 1/q, and we can interpret this by saying that the "events" of being divisible by p and q are independent. This holds, of course, for any number of primes, and we can say using a picturesque but not very precise language, that the primes play a game of chance! This simple, nearly trivial, observation is the beginning of a new development which links in a significant way number theory on the one hand and probability theory on the other."

M. Kac, Statistical Independence in Probability, Analysis and Number Theory. (Wiley, 1959)

"Tables of prime numbers display a chaotic aspect whose apparent disorder somewhat resembles classical random models arising, for example, from physical phenomena. And here is exactly the purpose of this little book: to describe, and then try to understand, how a sequence so precisely determined as that of prime numbers can incorporate so great a share of randomness.

Let us take this point a little further. Total randomness, chaos, is infinite complexity. Besides, the complexity of an integer grows with its size: is the number 26972593-1 prime? In the neighbourhood of infinity, the sequence of integers, and therefore that of prime numbers, contains randomness. Current research in modern analytic number theory tries to account for this aspect."

G. Tenenbaum and M. Mendès France, from The Prime Numbers and Their Distribution (AMS, 2000) page xii

"Given the sporadic, random-like quality of the primes, it is quite surprising how much can be proved about them. Interestingly, theorems about the primes are usually proved by exploiting this seeming randomness...

Much research on prime numbers has this sort of flavour. Your first devise a probabilistic model for the primes - that is, you pretend to yourself that they have been selected according to some random procedure. Next, you work out what would be true if the primes really were generated randomly. That allows you to guess the answers to many questions. Finally, you try to show that the model is realistic enough for you guesses to be approximately correct...

It is interesting that the probabilistic model is a model not of a physical phenomenon, but of another piece of mathematics. Although the prime numbers are rigidly determined, they somehow feel like experimental data. Once we regard them that way, it becomes tempting to devise simplified models that allow us to predict what the answers to certain probabilistic questions are likely to be. And such models have indeed sometimes led people to proofs valid for the primes themselves."

T. Gowers, Mathematics: A Very Short Introduction (Oxford Univ. Press, 2002) p.120-121

"I think that it's just a completely outrageous conjecture to make in the first place that probability should have anything to do with the primes."

T. Gowers, from his keynote talk "The Importance of Mathematics" at The Clay Mathematics Institute Millenium Meeting (2000) [see here, at about 1:40]

"Prime numbers present mathematicians with one of the strangest tensions in their subject. On the one hand a number is either prime or it isn't. No flip of a coin will suddenly make a number divisible by some smaller number. Yet there is no denying that the list of primes looks like a randomly chosen sequence of numbers. Physicists have grown used to the idea that a quantum die decides the fate of the universe, randomly choosing at each throw where scientists will find matter. But it is something of an embarrassment to have to admit that these fundamental numbers on which mathematics is based appear to have been laid out by Nature flipping a coin, deciding at each toss the fate of each number. Randomness and chaos are anathema to the mathematician."

M. du Sautoy, The Music of the Primes (Fourth Estate, 2003) p.6

"Probability is not a notion of pure mathematics, but of physics or philosophy."

G.H Hardy and J.E. Littlewood, "Some problems of 'partitio numerorum:' III: on the expression of a number as a sum of primes" Acta Mathematica 44 (1922)

"One of the remarkable aspects of the distribution of prime numbers is their tendency to exhibit global regularity and local irregularity. The prime numbers behave like the 'ideal gases' which physicists are so fond of. Considered from an external point of view, the distribution is - in broad terms - deterministic, but as soon as we try to describe the situation at a given point, statistical fluctuations occur as in a game of chance where it is known that on average the heads will match the tail but where, at any one moment, the next throw cannot be predicted. Prime numbers try to occupy all the room available (meaning that they behave as randomly as possible), given that they need to be compatible with the drastic constraint imposed on them, namely to generate the ultra-regular sequence of integers.

This idea underpins the majority of conjectures concerning prime numbers: everything which is not trivially forbidden should actually happen..."

G. Tenenbaum and M. Mendès France, from The Prime Numbers and Their Distribution (AMS, 2000) page 51

"In investigations concerning the asymptotic properties of arithmetic functions, it is often possible to make an interesting use of probability arguments. If, e.g., we are interested in the distribution of a given sequence S of integers, we then consider S as a member of an infinite class C of sequences, which may be concretely interpreted as the possible realizations of some game of chance. It is then in many cases possible to prove that, with a probability equal to 1, a certain relation R holds in C, i.e., that in a definite mathematical sense, 'almost all' sequences of C satisfy R. Of course we cannot, in general, conclude that R holds for the particular sequence S, but results suggested in this way may sometimes afterwards be rigorously proved by other methods."

H. Cramér, "On the order of magnitude of the difference between consecutive prime numbers", Acta Arithmetica 2 (1937) 23-46.

In an homage to the revered mathematician Paul Erdös, who died September 20, 1996, D. Mackenzie mentioned a theory Erdös published in 1940 with M. Kac. This theory states that a plot of the number of prime factors of very large numbers forms a bell curve - almost as if these numbers were "choosing" their prime factors at random. Alluding to an assertion Einstein is said to have made, Carl Pomerance has suggested that Erdös might have said something like this:

"God may not play dice with the universe, but something strange is going on with the prime numbers."

D. Mackenzie, "Homage to an Itinerant Master", Science 275 (1997) 759 (this mistakenly attributes the quotes to Erdös himself)


historical background - H. Cramér's contribution to the field

Although Erdös, Kac, Wintner and Kubilius are generally credited with the founding of probabilistic number theory, the following (earlier) work of Harald Cramér should not be overlooked:

H. Cramér, "Prime numbers and probability", Skand. Mat.-Kongr. 8 (1935) 107-115.

A. Granville, "Harald Cramér and the distribution of prime numbers", Scandanavian Actuarial Journal 1 (1995), 12-28. [an excellent survey of Cramér's work in this area]

The following is an excerpt from the chapter "Stochastic Distribution of Prime Numbers" of G. Tenenbaum and M. Mendès France's wonderful little book The Prime Numbers and Their Distribution (AMS, 2000):

"The most natural questions about prime numbers are often quite hard to answer. Furthermore, it is even rather difficult, in certain cases, to make a reasonable guess for what the exact answer might be. Having studied the distribution of prime numbers since the twenties, Cramér proposed at the end of the thirties a simple and fascinating method for producing conjectures concerning primes: the sequence of prime numbers behaves like a random sequence with the same growth constraint.

Let us formalize a little. The prime number theorem suggests that the probability that an integer of size n be a prime is close to 1/log n. Denote by {Xn} (where n = 2,3,4,...) a sequence of independent random variables taking values 0 and 1 with the

probability(Xn = 1) = 1/log n       (n > 2)

Then the random sequence

S := {n > 1: Xn = 1}

constitutes, according to Cramér, a stochastic model of the sequence of primes.

In more intuitive terms, we may consider the following mechanism. Arrange, in an infinite (or very long) sequence, urns U3, U4, etc., with the property that the urn Un contains one white ball and about log n black balls. Now choose a ball at random from each urn and assign the integer n to the sequence S if the ball taken from Un is white. The sequence S models the sequence of primes. Of course, for each particular drawing the sequence S will possess specific properties which will distinguish it significantly from the sequence P of all prime numbers. However, if a property turns up sufficiently frequently to occur almost surely when the drawings are repeated, we conjecture with Cramér that it is also shared by P."

A clearly-written elementary introduction to Cramér's ideas is contained in the following article:

D.L. Snell, "Chance in the Primes, Part II", Chance News 11.02


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 Tz 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 Tz [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] one of a family of strings on which there is a probability measure..."


(approximately) chronological bibliography

P. Doyle's English translation of von Sternach's 1896 treatise on number theoretical random walks, associated with this introductory article on the subject (scroll down to the section "Probability and the Riemann Hypothesis").

P. Erdös and A. Wintner, "Additive arithmetic functions and statistical independence", American Journal of Mathematics 61 (1939) 713-722.

P. Erdös and M. Kac, "The Gaussian law of errors in the theory of additive number theoretic functions", American Journal of Mathematics 62 (1940) 738-742.

notes on the Erdös-Kac theorem

H. Riesel, "The Erdös-Kac Theorem", in Prime Numbers and Computer Methods for Factorization, 2nd ed. (Birkhäuser, 1994) pp. 158-159

"The Prime Number Theorem obtained by statistical methods" - a heuristic argument from What is Mathematics? by R. Courant and H. Robbins (1941)

M. Kac, "Probability methods in some problems of analysis and number theory", Bulletin of the AMS (1949) 641-665.

A. Rényi, "On the density of certain sequences of integers", Publ. Inst. Math. Belgrade 8 (1955) 157-162.

I.P. Kubilius, "Probability methods in number theory" (in Russian), Usp. Mat. Nauk 68 (1956) 31-66.

A. Rényi and P. Turán, "On a theorem of Erdös-Kac", Acta. Arith. 4 (1958) 71-84.

D. Hawkins, "The random sieve", Mathematics Magazine 31(1958) 1-3.

D. Hawkins, "Random sieves, II", Journal of Number Theory 6 (1974) 192-200.

A. Denjoy, "Probabilites confirmant l'hypothese de Riemann sur les zeros de zeta(s), C.R. Acad. Sci. Paris 259 (1964) 3143-3145.

Notes on Denjoy's probabilistic interpretation of the Riemann Hypothesis from H. Edwards book Riemann's Zeta Function

I.J. Good and R.F. Churchhouse, "The Riemann hypothesis and pseudorandom features of the Möbius sequence", Mathematics of Computation 22 (1968) 857-864

[abstract:] "A study of the cumulative sums of the Möbius function on the Atlas Computer of the Science Research Council has revealed certain statistical properties which lead the authors to make a number of conjectures. One of these is that any conjecture of the Mertens type, viz.

$|M(N)| = |\sum_{n=1}^{N}\mu(n)| \lt k(\sqrt(N))$

where $k$ is a positive constant, is false, and indeed, the authors conjecture that

$\Lim \sup {M(x)(x \log \log x)^{-1/2}} = \sqrt{(12)/\pi}$

S. Golomb, "A class of probability distributions on the integers", Journal of Number Theory 2 (1970) 189-192. 

P. Billingsley, "Prime numbers and Brownian motion", American Mathematical Monthly 80 (1973) 1099. 

W. Neudecker and D. Williams, "The 'Riemann Hypothesis' for the Hawkins random sieve" Compositio Mathematica 29 (1974) 197-200.

P. Nanopoulos, "Loi de Dirichlet sur N* et pseudo-probabilites", C.R. Acad. Sci. Paris Ser. A-B 280 (22) (1975) A1543-A1546. 

C.C. Heyde, "On asymptotic behavior for the Hawkins random sieve", Proc. AMS 56 (1976) 277-280

[abstract:] "This paper is concerned with the Hawkins random sieve which is a probabilistic analogue of the sieve of Eratosthenes. Analogues of the prime number theorem and Mertens' theorem have previously been obtained for this sieve by classical probabilistic methods. In the present paper, sharper results akin to the Riemann hypothesis are obtained by a more elegant martingale approach."

P. Diaconis, F. Mosteller and H. Onishi, "Second order terms for the variances and covariances of the number of prime factors - including the square free case", Journal of Number Theory 9 (1977) 187-202.

C.C. Heyde, "A log log improvement to the Riemann Hypothesis for the Hawkins random sieve" Ann. Prob. 6 no.5 (1978) 870-875

[abstract:] "This paper is concerned with the Hawkins random sieve which is a probabilistic analouge of the sieve of Eratosthenes. Analogues of the prime number theorem, Mertens' theorem and the Riemann hypothesis have previously been established for the Hawkins sieve. In the present paper we give a more delicate analysis using iterated logarithm results for both mantingales and tail sums of martingale differences to deduce a considerably improved log log replacement for the Riemann hypothesis result."

J. Kubilius, Probabilistic Methods in the Theory of Numbers (AMS, 1978)

P.D.T.A. Elliot, Probabilistic Number Theory I: Mean-value Theorems, Grundlehren der Mathematischen Wissenschaften 239 (Springer, 1979)

P.D.T.A. Elliot, Probabilistic Number Theory II: Central Limit Theorems, Grundlehren der Mathematischen Wissenschaften 240 (Springer, 1980)

R.O. Rabin, "Probabilistic algorithm for testing primality", Journal of Number Theory 12 (1980) 128-138.

M. Shlesinger, "On the Riemann hypothesis: a fractal random walk approach", Physica A 138 (1986) 310-319

[abstract:] "In his investigation of the distribution of prime numbers Riemann, in 1859, introduced the zeta function with a complex argument. His analysis led him to hypothesize that all the complex zeros of the zeta function lie on a vertical line in the complex plane. The proof or disproof of this hypothesis has been a famous outstanding problem in mathematics. We are able to recast Riemann's Hypothesis into a probabilistic framework connected to the fractal behavior of a lattice random walk. Fractal random walks were introduced by P. Levy, and in the continuum are called Levy flights. For one particular lattice version of a Levy flight we show the connection to Weierstrass' continuous but nowhere differentiable function. For a different lattice version, using a Mellin transform analysis, we show how the zeroes of the zeta function become the singularities of a complex integrand which governs the behavior of a fractal random walk. The laws of probability place restrictions on the locations of the zeroes of the zeta function. No inconsistencies with probability theory are found if the Riemann Hypothesis is false."

V.K. Murty and M.R. Murty, "An analog of the Erdös-Kac theorem for Fourier coefficients of modular forms", Indian Journal of Pure and Applied Mathematics 15 (10) (1984) 1090-1101.

L. Smith and P. Diaconis, "Honest Bernoulli excursions", Journal of Applied Probability 25 (1988) 464-477

J.V. Armitage, "The Riemann Hypothesis and the Hamiltonian of a quantum mechanical system" (section 5: "A random walk approximation to the Riemann Hypothesis"), from Number Theory and Dynamical Systems, eds. M.M. Dodson and J.A.G. Vickers (LMS Lecture Notes, series 134, Cambridge University Press), 153-172. 

"The connection between random walks and Brownian motion is well-known and so also the connection with the Schrödinger equation, on replacing 'time' with 'imaginary time'. In this section we use a random walk approach to the Ornstein-Uhlenbeck process (or the Fokker-Planck equation) to exhibit a polynomial whose zeros, under a suitable limiting process, ought to be the zeros of the Riemann zeta-function." 

D. Williams, "Brownian motion and the Riemann zeta-function", from Disorder in Physical Systems (Clarendon Press, 1990) 361-372.

J.L. Lucio and Y. Meurice, "Asymptotic properties of random walks on p-adic spaces", University of Iowa Preprint 90-33, 1-5 (1990)

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."

K.S. Alexander, K. Baclawski and G.-C. Rota, "A stochastic interpretation of the Riemann zeta function", Proceedings of the National Academy of Sciences USA 90 (1993) 697-699

[abstract:] "We give a stochastic process for which the terms of the Riemann zeta function occur as the probability distributions of the elementary random variables of the process."

S. Albeverio and W. Karwowski, "A Random Walk on p-Adics - the Generator and Its Spectrum", Stochastic Processes and their Applications 53 (1994) 1-22.

G. Tenenbaum, Introduction to Analytic and Probabilistic Number Theory, Cambridge Studies in Advanced Mathematics 46 (C.U.P., 1995)

N. Boston, "A probabilistic generalization of the Riemann zeta function", Analytic Number Theory, Vol. 1, Progr. Math. 138, (Birkhauser, 1996) 155-162.

P. Joshi and S. Chakraborty, "Moments of Cauchy order statistics via Riemann zeta functions", from Statistical Theory and Applications (editors H.N. Nagaraja, et al.) 111-127 (Springer 1996)

C. Calude, P. Hertling and B. Khoussainov, "Do the zeros of Riemann's zeta-function form a random sequence?", Bulletin of the EATCS 62 (1997) 199-207

J. Chang and Y. Peres, "Ladder Heights, Gaussian random walks, and the Riemann zeta function", Annals of Probability, 25 (1997) 787-802.

[abstract:] "Let $\{S_n:n\ge 0\}$ be a random walk having normally distributed increments with mean $\theta$ and variance 1, and let $\tau$ be the time at which the random walk first takes a positive value, so that $S_\tau$ is the first ladder height. Then the expected value $E_\theta S_\tau$, originally defined for $\theta>0$, may be extended to be an analytic function of the complex variable $\theta$ throughout the complex plane, with the exception of certain branch point singularities. In particular, the coefficients in a Taylor expansion about $\theta=0$ may be written explicitly as simple expressions involving the Riemann zeta function. Previously only the first coefficient of the series developed here was known; this term has been used extensively in developing approximations for boundary crossing problems for Gaussian random walks. Knowledge of the complete series makes more refined results possible; we apply it to derive asymptotics for boundary crossing probabilities and the limiting expected overshoot."

P. Biane, J. Pitman, and M. Yor, "Probability laws related to the Jacobi theta and Riemann zeta functions, and Brownian excursions", Bulletin of the AMS 38 (2001) 435-465.

J. Pitman and M. Yor, "Infinitely divisible laws associated with hyperbolic functions", Canadian Journal of Mathematics 55 no.2 (2003) 292-330

[abstract:] "The infinitely divisible distributions of positive random variables Ct, St and Tt with Laplace transforms in x

(1/(cosh(2x)1/2))t, ((2x)1/2/sinh((2x)1/2))t, and (tanh((2x)1/2)/(2x)1/2)t

respectively are characterized for various t > 0 in a number of different ways: by simple relations between their moments and cumulants, by corresponding relations between the distributions and their Levy measures, by recursions for their Mellin transforms, and by differential equaitons satisfied by their Laplace transforms.

Some of these results are interpreted probabilistically via known appearances of these distributions for t = 1 or 2 in the description of the laws of various functionals of Brownian motion and Bessel processes, such as the heights and lengths of excursions of a one-dimensional brownian motion. The distributions of C1 and S2 are also known to appear in the Mellin representations of two important functions in analytic number theory, the Riemann zeta function and the Dirichlet L-function associated with the quadratic character modulo 4.

Related families of infinitely divisible laws, including the gamma, logistic and generalised hyperbolic secant distributions, are derived from St and Ct by operations such as Brownian subordinations, exponential tilting, and weak limits, and characterized in various ways."

Marc Yor is acknowledged for "fascinating discussions on the connection between [a theorem in the following paper] and planar Brownian motion" here:

C. Hughes, J. Keating, and N. O'Connell, "On the Characteristic Polynomial of a Random Unitary Matrix", Communications in Mathematical Physics 220 (2001) 429-451

S. Asmussen, P. Glynn and J. Pitman, "Discretization error in simulation of one-dimensional reflecting Brownian motion", Annals of Applied Probability 5 (1995) 875-896

Analytic and Probabilistic Methods in Number Theory - New Trends in Probability and Statistics, Volume 4 (VSP, 1997)

C. M. Bender, S. Boettcher, L. R. Mead, "Random walks in noninteger dimension", Journal of Mathematical Physics 35 (1994) 368-388

"One can define a random walk on a hypercubic lattice in a space of integer dimension D... In this paper we propose a random walk which gives acceptable probabilities for all real values of D. This D-dimensional random walk is defined on a rotationally-symmetric geometry consisting of concentric spheres. We give the exact result for the probability of returning to the origin for all values of D in terms of the Riemann zeta function. This result has a number-theoretic interpretation." 

M. Mystkowski, "Random walk on p-adics with non-zero killing part", Reports on Mathematical Physics 34 (1994) 133-141

S. Albeverio and W. Karwowski, "A random walk on p-adics - the generator and its spectrum", Stochastic Processes and their Applications, 53 (1994) 1-22

S. Albeverio, W. Karwowski and X. Zhao, "Asymptotics and spectral results for random walks on p-adics", Stochastic Processes and their Applications, 83 (1999)

M. Wolf, "Random walk on the prime numbers", Physica A 250 (1998) 335. 

I. Vardi, "Deterministic percolation", Communications in Mathematical Physics 207 (1999) 43-66

[excerpt from introduction:] "...percolation theory has been of great interest in physics, as it is one of the simplest models to exhibit phase transitions. In this paper, I will examine how questions of percolation theory can be posed in a deterministic setting. Thus deterministic percolation is the study of unbounded walks on a single subset of a graph, e.g., defined by number theoretic conditions. This might be of interest in physics and probability theory as it studies percolation in a deterministic setting and in number theory where it can be interpreted as studying the disorder inherent in the natural numbers."

I. Vardi, "Prime percolation", Experimental Mathematics 7 (1998) 275-288

[abstract:] "This paper examines the question of whether there is an unbounded walk of bounded step size along Gaussian primes. Percolation theory predicts that for a low enough density of random Gaussian integers no walk exists, which suggests that no such walk exists along prime numbers, since they have arbitrarily small density over large enough regions. In analogy with the Cramér conjecture, I construct a random model of Gaussian primes and show that an unbounded walk of step size $k\sqrt{\log|z|}$ at $z$ exists with probability 1 if $k \gt \sqrt{2\pi\lambda_{c}}$ and does not exist with probability 1 if $k \lt \sqrt{2\pi\lambda_{c}}$ where $\lambda_{c}$ ~ 0.35 is a constant in continuum percolation, and so conjecture that the critical step size for Gaussian primes is also $\sqrt{2\pi\lambda_{c}\log|z|}$.

J.-F. Burnol, "The Explicit Formula and a propagator" (preprint, 09/98)

[abstract:] "I give a new derivation of the Explicit Formula for the general number field K, which treats all primes in exactly the same way, whether they are discrete or archimedean, and also ramified or not. In another token, I advance a probabilistic interpretation of Weil's positivity criterion, as opposed to the usual geometrical analogies or goals. But in the end, I argue that the new formulation of the Explicit Formula signals a specific link with Quantum Fields, as opposed to the Hilbert-Pólya operator idea (which leads rather to Quantum Mechanics)."

[excerpt:] "To express his positivity criterion Weil uses conventions slightly distinct from ours. He moves the local terms to be together with the poles, and makes a shift of 1/2 in the Mellin transform. In this way he gets a distribution C and translates the Riemann Hypothesis into a positivity criterion:

$C(F \star F^{\tau}) \geq 0$

for an arbitrary test-function F on C...In the function field case $C(F \star F^{\tau})$ can be given a geometric interpretation as an intersection number of cycles on an algebraic surface, and the positivity follows from the Hodge Index Theorem...

But another interpretation is possible that does not seem to have been pushed forward so far. To prove that a number is non-negative it is enough to exhibit it as the variance of a random variable. In our case this means taht there should be a generalized, stationary, zero mean, stochastic process with C as "time" whose covariance would be C. That is we have a probability measure $mu$ on the distribgutions on the classes of ideles...

If such a probability measure $mu$ could be constructed, corresponding to an 'arithmetic stochastic process', then the Riemann Hypothesis would follow of occurs."

C. Donati-Martin, Zhan Shi and M. Yor, "The joint law of the last zeros of Brownian motion and of its Lévy transform", Ergodic Theory and Dynamical Systems 20 (2000) 709-725.

C. Castro (Perelman), "p-Adic stochastic dynamics, supersymmetry and the Riemann conjecture" (preprint 01/01)

"Supersymmetry, p-adic stochastic dynamics, Brownian motion, Fokker-Planck equation, Langevin equation, prime number random distribution, random matrices, p-adic fractal strings, the adelic condition, etc. ..are all deeply interconnected in this paper."

C. Castro and J. Mahecha, "Fractal supersymmetric quantum mechanics, geometric probability and the Riemann Hypothesis", International Journal of Geometric Methods in Modern Physics 1 no. 6 (2004) 751-793

[abstract:] "The Riemann Hypothesis (RH) states that the nontrivial zeros of the Riemann zeta-function are of the form $s = 1/2 + i\lambda_{n}$. Earlier work on the RH based on Supersymmetric QM, whose potential was related to the Gauss-Jacobi theta series, allows to provide the proper framework to construct the well defined algorithm to compute the probability to find a zero (an infinity of zeros) in the critical line. Geometric Probability Theory furnishes the answer to the very difficult question whether the probability that the RH is true is indeed equal to unity or not. To test the validity of this Geometric Probabilistic framework to compute the probability if the RH is true, we apply it directly to the hyperbolic sine function sinh(s) case which obeys a trivial analog of the RH. Its zeros are equally spaced in the imaginary axis $s_n = 0 + in\pi$. The Geometric Probability to find a zero (and an infinity of zeros) in the imaginary axis is exactly unity. We proceed with a fractal supersymmetric quantum mechanical (SUSY-QM) model to implement the Hilbert-Pólya proposal to prove the RH by postulating a Hermitian operator that reproduces all the $\lambda_n$'s for its spectrum. Quantum inverse scattering methods related to a fractal potential given by a Weierstrass function (continuous but nowhere differentiable) are applied to the analog of the fractal analog of the CBC (Comtet-Bandrauk-Campbell) formula in SUSY QM. It requires using suitable fractal derivatives and integrals of irrational order whose parameter $\beta$ is one-half the fractal dimension (D = 1.5) of the Weierstrass function. An ordinary SUSY-QM oscillator is also constructed whose eigenvalues are of the form $\lambda_n = n\pi$ and which coincide which the imaginary parts of the zeros of the function sinh(s). Finally, we discuss the relationship to the theory of 1/f noise."

P.F. Kelly and T. Pilling, "Physically inspired analysis of prime number constellations" (preprint, 08/01)

"We adopt a physically motivated empirical approach to the characterisation of the distributions of twin and triplet primes within the set of primes, rather than in the set of all natural numbers. Remarkably, the occurrences of twins or triplets in any finite sequence of primes are like fixed-probability random events. The respective probabilities are not constant, but instead depend on the length of the sequence in ways that we have been able to parameterise. For twins the "decay constant'' decreases as the reciprocal of the logarithm of the length of the sequence, whereas for triplets the falloff is faster: decreasing as the square of the reciprocal of the logarithm of the number of primes. The manner of the decrease is consistent with the Hardy-Littlewood Conjectures, developed using purely number theoretic tools of analysis."

A.N. Kochubei, Pseudo-differential Equations and Stochastics over non-Archimedian Fields (Marcel Dekker, 2001)

(from publisher's description) "This state-of-the-art reference provides comprehensive coverage of the most recent developments in the theory of non-Archimedean pseudo-differential equations and its application to stochastics and mathematical physics offering current methods of construction for stochastic processes on the field of p-adic numbers and related structures. Develops a new theory for parabolic equations over non-Archimedean fields in relation to Markov processes!"

D.L. Snell, "Chance in the Primes, Part II", Chance News 11.02

This clearly-written elementary article provides examples of "how chance has been used in the study of prime numbers". Probabilistic 'proof' of the Prime Number Theorem, Harald Cramér's 'random primes', prime-based random walks, sieving, cryptography and the Möbius function are all discussed.

notes on the probabilities associated with "visible point" lattice problems and the Riemann zeta function

F. Todor, "A probability interpretation for the Zeta function of Riemann and some technical regularization for spectral representation" (paper presented to the 11th Conference on Applied and Industrial Mathematics, Romania, May 2003)

G. Chaitin, "Some thoughts on the Riemann Hypothesis (preprint, 06/03)

[excerpt:] "I have always had an interest in probabilistic methods in elementary number theory. This was one of the things that inspired me to come up with my definition of algorithmic randomness and to find algorithmic randomness in arithmetic in connection with diophantine equation."

Chaitin goes on to recommend the following:

G. Pólya, "Heuristic reasoning in the theory of numbers", American Mathematical Monthly 66 (1957) 375-384

B.M. Hambly and M.L. Lapidus, "Random fractal strings: their zeta functions, complex dimensions and spectral asymptotics" (preprint 02/04)

[abstract:] "In this paper a string is a sequence of positive non-increasing real numbers which sums to one. For our purposes a fractal string is a string formed from the lengths of removed sub-intervals created by a recursive decomposition of the unit interval. By using the so called complex dimensions of the string, the poles of an associated zeta function, it is possible to obtain detailed information about the behaviour of the asymptotic properties of the string. We consider random versions of fractal strings. We show that using a random recursive self-similar construction it is possible to obtain similar results to those for deterministic self-similar strings. In the case of strings generated by the excursions of stable subordinators, we show that the complex dimensions can only lie on the real line. The results allow us to discuss the geometric and spectral asymptotics of one-dimensional domains with random fractal boundary."

S.N. Evangelou, D.E. Katsanos, "Quantum correlations from Brownian diffusion of chaotic level-spacings" (preprint 04/04)

[abstract:] "Quantum chaos is linked to Brownian diffusion of the underlying quantum energy level-spacing sequences. The level-spacings viewed as functions of their order execute random walks which imply uncorrelated random increments of the level-spacings while the integrability to chaos transition becomes a change from Poisson to Gauss statistics for the level-spacing increments. This universal nature of quantum chaotic spectral correlations is numerically demonstrated for eigenvalues from random tight binding lattices and for zeros of the Riemann zeta function."

G. Corso, "Families and clustering in a natural numbers network", Phys. Rev. E 69 (2004)

"We develop a network in which the natural numbers are the vertices. The decomposition of natural numbers by prime numbers is used to establish the connections. We perform data collapse and show that the degree distribution of these networks scales linearly with the number of vertices. We explore the families of vertices in connection with prime numbers decomposition. We compare the average distance of the network and the clustering coefficient with the distance and clustering coefficient of the corresponding random graph. In case we set connections among vertices each time the numbers share a common prime number the network has properties similar to a random graph. If the criterion for establishing links becomes more selective, only prime numbers greater than pl are used to establish links, where the network has high clustering coefficient."

B. Green and T. Tao, "The primes contain arbitrarily long arithmetic progressions" (preprint 04/04)

[abstract:] "We prove that there are arbitrarily long arithmetic progressions of primes. There are three major ingredients. The first is Szemerédi's theorem, which asserts that any subset of the integers of positive density contains progressions of arbitrary length. The second, which is the main new ingredient of this paper, is a certain transference principle. This allows us to deduce from Szemerédi's theorem that any subset of a sufficiently pseudorandom set of positive relative density contains progressions of arbitrary length. The third ingredient is a recent result of Goldston and Yildirim. Using this, one may place the primes inside a pseudorandom set of 'almost primes' with positive relative density."

[from proof outline, p.4] "Perhaps surprisingly for a result about primes, our paper has at least as much in common with the ergodic-theoretic approach as it does with the harmonic analysis approach of Gowers. We will use a language which suggests this close connection, without actually relying explicitly on any ergodic theoretical concepts".

T. Tao, "Obstructions to uniformity, and arithmetic patterns in the primes" (preprint 05/05, submitted for special edition of Quarterly J. Pure Appl. Math. in honour of John Coates)

[abstract:] "In this expository article, we describe the recent approach, motivated by ergodic theory, towards detecting arithmetic patterns in the primes, and in particular establishing that the primes contain arbitrarily long arithmetic progressions. One of the driving philosophies is to identify precisely what the obstructions could be that prevent the primes (or any other set) from behaving 'randomly', and then either show that the obstructions do not actually occur, or else convert the obstructions into usable structural information on the primes."

B. Kra, "The Green-Tao Theorem on arithmetic progressions in the primes: an ergodic point of view", Bull. Amer. Math. Soc. 43 (2006), 3-23

[abstract:] "A long-standing and almost folkloric conjecture is that the primes contain arbitrarily long arithmetic progressions. Until recently, the only progress on this conjecture was due to van der Corput, who showed in 1939 that there are infinitely many triples of primes in arithmetic progression. In an amazing fusion of methods from analytic number theory and ergodic theory, Ben Green and Terence Tao showed that for any positive integer k, there exist infinitely many arithmetic progressions of length k consisting only of prime numbers. This is an introduction to some of the ideas in the proof, concentrating on the connections to ergodic theory."

Z. Rudnick, "A Central Limit Theorem for the spectrum of the modular domain" (to appear in Annales Henri Poincaré)

[abstract:] "We study the fluctuations in the discrete spectrum of the hyperbolic Laplacian for the modular domain using smooth counting functions. We show that in a certain regime, these have Gaussian fluctuations."

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.

[abstract:] "The effect of Richard T. Cox's contribution to probability theory was to generalize Boolean implication among logical statements to degrees of implication, which are manipulated using rules derived from consistency with Boolean algebra. These rules are known as the sum rule, the product rule and Bayes' Theorem, and the measure resulting from this generalization is probability. In this paper, I will describe how Cox's technique can be further generalized to include other algebras and hence other problems in science and mathematics. The result is a methodology that can be used to generalize an algebra to a calculus by relying on consistency with order theory to derive the laws of the calculus. My goals are to clear up the mysteries as to why the same basic structure found in probability theory appears in other contexts, to better understand the foundations of probability theory, and to extend these ideas to other areas by developing new mathematics and new physics. The relevance of this methodology will be demonstrated using examples from 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."

Probability and Number Theory - Kanazawa 2005, eds. S. Akiyama, et al. (Mathematical Society of Japan, 2008)

[publisher's description:] "This volume is the proceedings of the International Conference on Probability and Number Theory held at Kanazawa, Japan, in June 2005 and includes several survey articles on probabilistic number theory and research papers on various recent topics around the border area between probability theory and number theory. This volume is useful for all researchers and graduate students who are interested in probability theory and number theory."

A. Kubasiak, J. Korbicz, J. Zakrzewski, M. Lewenstein, "Fermi-Dirac statistics and the number theory" (preprint 07/05)

[abstract:] "We relate the Fermi-Dirac statistics of an ideal Fermi gas in a harmonic trap to partitions of given integers into distinct parts, studied in number theory. Using methods of quantum statistical physics we derive analytic expressions for cumulants of the probability distribution of the number of different partitions."

C.S. Calude and M. Stay, "Natural halting probabilities, partial randomness, and zeta functions" (preprint 01/06)

[abstract:] "We introduce the natural halting probability and the natural complexity of a Turing machine and we relate them to program-size complexity and Chaitin's halting probability. A classification of Turing machines according to their natural (Omega) halting probabilities is proposed: divergent, convergent and tuatara. We prove the existence of universal convergent and tuatara machines. Various results on randomness and partial randomness are proved. For example, we show that the natural halting probability of a universal tuatara machine is c.e. and random. A new type of partial randomness, asymptotic randomness, is introduced. Finally we show that in contrast to classical (algorithmic) randomness - which cannot be characterised in terms of plain complexity - various types of partial randomness admit such characterisations."

O. Shanker, "Zeroes of Riemann zeta function and Hurst exponent" (preprint 01/06)

[abstract:] "The theory underlying the location of the zeros of the Riemann zeta function is one of the most intriguing unsolved problems. It is interesting to physicists because of the Hilbert-Pólya Conjecture, that the non-trivial zeros of the zeta function correspond to the eigenvalues of some positive operator. Since there is no proof yet for this conjecture, it is important to study the properties of the locations of the zeroes empirically using a variety of methods. In this paper we use the rescaled range analysis to study the spacings between successive zeroes. We find that for large orders of the zeroes the spacings seem to have a Hurst exponent of about 0.095. This implies that the distribution has a high fractal dimension, and shows a lot of detailed structure. The distribution appears to be of the anti-persistent fractional Brownian motion type, with a significant degree of anti-persistence."

Wang Liang and Huang Yan, "Pseudo Random test of prime numbers" (preprint 03/2006)

[abstract:] "The prime numbers look like a randomly chosen sequence of natural numbers, but there is still no strict theory to determine 'Randomness'. In these years, cryptography has developed a battery of statistical tests for randomness. In this paper, we just apply these methods to study the distribution of primes. Here the binary sequence constructed by second difference of primes is used as samples. We find this sequence can't reach all the 'random standard' of FIPS 140-1/2, but still show obvious random feature. The interesting self-similarity is also observed in this sequence. These results add the evidence that prime numbers is a chaos system."

M.Katori, M. Izumi and N. Kobayashi, "Two Bessel bridges conditioned never to collide, double Dirichlet series, and Jacobi theta function" (preprint 11/07)

[abstract:] "It is known that the moments of the maximum value of a one-dimensional conditional Brownian motion, the three-dimensional Bessel bridge with duration 1 started from the origin, are expressed using the Riemann zeta function. We consider a system of two Bessel bridges, in which noncolliding condition is imposed. We show that the moments of the maximum value is then expressed using the double Dirichlet series, or using the integrals of products of the Jacobi theta functions and its derivatives. Since the present system will be provided as a diffusion scaling limit of a version of vicious walker model, the ensemble of 2-watermelons with a wall, the dominant terms in long-time asymptotics of moments of height of 2-watermelons are completely determined, for which only the first moment, i.e. the average height, was recently studied by Fulmek by a method of enumerative combinatorics."

L. Lacasa, B. Luque, O. Miramontes, "Phase transition and computational complexity in a stochastic prime number generator" (preprint 12/2007)

[abstract:] "We introduce a prime number generator in the form of a stochastic algorithm. The character of such algorithm gives rise to a continuous phase transition which distinguishes a phase where the algorithm is able to reduce the whole system of numbers into primes and a phase where the system reaches a frozen state with low prime density. In this paper we firstly pretend to give a broad characterization of this phase transition, both in terms of analytical and numerical analysis. Critical exponents are calculated, and data collapse is provided. Further on we redefine the model as a search problem, fitting it in the hallmark of computational complexity theory. We suggest that the system belongs to the class NP. The computational cost is maximal around the threshold, as common in many algorithmic phase transitions, revealing the presence of an easy-hard-easy pattern. We finally relate the nature of the phase transition to an average-case classification of the problem."

Austrian National Research: Network Analytic Combinatorics and Probabilistic Number Theory (2006-2008)

E. Kowalski, The Large Sieve and its Applications, Arithmetic Geometry, Random Walks and Discrete Groups, Cambridge Tracts in Mathematics 175 (CUP, 2008)

N.Kobayashi, M. Izumi, M. Katori, "Maximum distributions of noncolliding Bessel bridges" (preprint 08/2008)

[abstract:] "The one-dimensional Brownian motion starting from the origin at time $t=0$, conditioned to return to the origin at time $t=1$ and to stay positive during time interval $0 < t < 1$, is called the Bessel bridge with duration 1. We consider the $N$-particle system of such Bessel bridges conditioned never to collide with each other in $0 < t < 1$, which is the continuum limit of the vicious walk model in watermelon configuration with a wall. Distributions of maximum-values of paths attained in the time interval $t \in (0,1)$ are studied to characterize the statistics of random patterns of the repulsive paths on the spatio-temporal plane. For the outermost path, the distribution function of maximum value is exactly determined for general $N$. We show that the present $N$-path system of noncolliding Bessel bridges is realized as the positive-eigenvalue process of the $2N \times 2N$ matrix-valued Brownian bridge in the symmetry class C. Using this fact computer simulations are performed and numerical results on the $N$-dependence of the maximum-value distributions of the inner paths are reported. The present work demonstrates that the extreme-value problems of noncolliding paths are related with the random matrix theory, representation theory of symmetry, and the number theory."

S. Torquato, A. Scardicchio, C.E. Zachary, "Point processes in arbitrary dimension from fermionic gases, random matrix theory, and number theory" (preprint 09/2008)

[abstract:] "It is well known that one can map certain properties of random matrices, fermionic gases, and zeros of the Riemann zeta function to a unique point process on the real line. Here we analytically provide exact generalizations of such a point process in $d$-dimensional Euclidean space for any $d$, which are special cases of determinantal processes. In particular, we obtain the $n$-particle correlation functions for any n, which completely specify the point processes. We also demonstrate that spin-polarized fermionic systems have these same $n$-particle correlation functions in each dimension. The point processes for any $d$ are shown to be hyperuniform. The latter result implies that the pair correlation function tends to unity for large pair distances with a decay rate that is controlled by the power law $r^{-(d+1)}$. We graphically display one- and two-dimensional realizations of the point processes in order to vividly reveal their "repulsive" nature. Indeed, we show that the point processes can be characterized by an effective "hard-core" diameter that grows like the square root of $d$. The nearest-neighbor distribution functions for these point processes are also evaluated and rigorously bounded. Among other results, this analysis reveals that the probability of finding a large spherical cavity of radius $r$ in dimension $d$ behaves like a Poisson point process but in dimension $d+1$ for large r and finite $d$. We also show that as $d$ increases, the point process behaves effectively like a sphere packing with a coverage fraction of space that is no denser than $1/2^d$."

N. Laskin, "Some Applications of the Fractional Poisson Probability Distribution" (preprint 12/2008)

[abstract:] "New physical and mathematical applications of recently invented fractional Poisson probability distribution have been presented. As a physical application, a new family of quantum coherent states have been introduced and studied. Mathematical applications are related to the number theory. We have developed fractional generalization of the Bell polynomials, the Bell numbers, and the Stirling numbers of the second kind. The fractional Bell polynomials appearance is natural if one evaluates the diagonal matrix element of the evolution operator in the basis of newly introduced quantum coherent states. The fractional Stirling numbers of the second kind have been introduced and applied to evaluate skewness and kurtosis of the fractional Poisson probability distribution function. A new representation of the Bernoulli numbers in terms of fractional Stirling numbers of the second kind has been found. In the limit case when the fractional Poisson distribution becomes the well-known Poisson probability distribution all of the above listed new developments and implementations turn into the well-known results of the quantum optics and the number theory."

B. Holdom, "Correlations, scale invariance and the Riemann Hypothesis" (preprint 03/2009)

[abstract:] "Negative correlations in the distribution of prime numbers are found to display a scale invariance. There are similarities and differences when compared to the scale invariant correlations of fractional Brownian motion. We conjecture that a violation of the Riemann hypothesis is equivalent to a breakdown of the scale invariance."

G. Garcia-Perez, M. Angeles Serrano and M. Boguna, "The complex architecture of primes and natural numbers" (preprint 02/2014)

[abstract:] "Natural numbers can be divided in two non-overlapping infinite sets, primes and composites, with composites factorizing into primes. Despite their apparent simplicity, the elucidation of the architecture of natural numbers with primes as building blocks remains elusive. Here, we propose a new approach to decoding the architecture of natural numbers based on complex networks and stochastic processes theory. We introduce a parameter-free non-Markovian dynamical model that naturally generates random primes and their relation with composite numbers with remarkable accuracy. Our model satisfies the prime number theorem as an emerging property and a refined version of Cram\'er's conjecture about the statistics of gaps between consecutive primes that seems closer to reality than the original Cram\'er's version. Regarding composites, the model helps us to derive the prime factors counting function, giving the probability of distinct prime factors for any integer. Probabilistic models like ours can help not only to conjecture but also to prove results about primes and the complex architecture of natural numbers."

B.M. Weon and Y. Kim, "Randomness in coin tosses and last digits of primes" (preprint 09/2014)

[abstract:] "Randomness is a central concept to statistics and physics. A new statistical analysis provides evidence that tossing coins and finding last digits of prime numbers are identical problems regarding equally likely outcomes. This analysis explains why randomness of equally likely outcomes is valid at large numbers."

G. França and A. LeClair, "On the validity of the Euler product inside the critical strip" (preprint 10/2014)

[abstract:] "The Euler Product Formula relates Riemann's zeta function $\zeta(s)$ to an infinite product over primes, and is known to be valid for $\Re(s) > 1$. We provide arguments that the formula is actually valid for $\Re(s) > 1/2$ and $\Im(s)\neq 0$ due to the conditional convergence of the infinite product in this regime. The argument relies on four ingredients: the prime number theorem, an Abel transform, a central limit theorem for the Random Walk of the Primes series $\sum_p\cos(t\log p)$, where $p$ is a prime number, and the Cauchy criterion for convergence. The significance of $\Re(s) > 1/2$ arises from the universality of the $N^{1/2}$ growth of fluctuations in various central limit theorems for independent and weakly dependent random processes, which are common in statistical physics for systems of size $N$. Numerical evidence of this surprising result is presented, and some of its consequences are discussed."

D. Ostrovsky, "On Riemann zeroes, lognormal multiplicative chaos, and Selberg integral" (preprint 06/2015)

[abstract:] "Rescaled Mellin-type transforms of the exponential functional of the Bourgade–Kuan–Rodgers statistic of Riemann zeroes are conjecturally related to the distribution of the total mass of the limit lognormal stochastic measure of Mandelbrot–Bacry–Muzy. The conjecture implies that a non-trivial, log-infinitely divisible probability distribution is associated with Riemann zeroes. For application, integral moments, covariance structure, multiscaling spectrum, and asymptotics associated with the exponential functional are computed in closed form using the known meromorphic extension of the Selberg integral."

A. Abdesselam, "Towards three-dimensional conformal probability" (preprint 10/2015)

[abstract:] "In this outline of a program, based on rigorous renormalization group theory, we introduce new definitions which allow one to formulate precise mathematical conjectures related to conformal invariance as studied by physicists in the area known as higher-dimensional conformal bootstrap which has developed at a breathtaking pace over the last five years. We also explore a second theme, intimately tied to conformal invariance for random distributions, which can be construed as a search for a very general first and second-quantized Kolmogorov–Chentsov Theorem. First-quantized refers to regularity of sample paths. Second-quantized refers to regularity of generalized functionals or Hida distributions and relates to the operator product expansion. Finally, we present a summary of progress made on a $p$-adic hierarchical model and point out possible connections to number theory."

A. LeClair, "Riemann Hypothesis and random walks: The zeta case" (preprint 01/2016)

[abstract:] "In previous work it was shown that if certain series based on sums over primes of non-principal Dirichlet characters have a conjectured random walk behavior, then the Euler product formula for its $L$-function is valid to the right of the critical line $\Re (s) > 1/2$, and the Riemann Hypothesis for this class of $L$-functions follows. Building on this work, here we propose how to extend this line of reasoning to the Riemann zeta function and other principal Dirichlet $L$-functions. We use our results to argue that $ S_\delta (t) \equiv \lim_{\delta \to 0^+} \dfrac{1}{\pi} \arg\zeta(\tfrac{1}{2}+ \delta + it) = O(1)$, and that it is nearly always on the principal branch. We conjecture that a 1-point correlation function of the Riemann zeros has a normal distribution. This leads to the construction of a probabilistic model for the zeros. Based on these results we describe a new algorithm for computing very high Riemann zeros as a kind of stochastic process, and we calculate the $10^{100}$-th zero to over 100 digits."

S.C.L. Srivastava, A. Lakshminarayan, S. Tomsovic and A. Bäcker, "Ordered level spacing probability densities" (preprint 10/2018)

[abstract:] "Spectral statistics of quantum systems have been studied in detail using the nearest neighbour level spacings, which for generic chaotic systems follows random matrix theory predictions. In this work, the probability density of the closest neighbour and farther neighbour spacings from a given level are introduced. Analytical predictions are derived using a $3\times 3$ matrix model. The closest neighbour density is generalized to the $k$-th closest neighbour spacing density, which allows for investigating long-range correlations. For larger $k$ the probability density of $k$-th closest neighbour spacings is well described by a Gaussian. Using these $k$-th closest neighbour spacings we propose the ratio of the closest neighbour to the second closest neighbour as an alternative to the ratio of successive spacings. For a Poissonian spectrum the density of the ratio is flat, whereas for the three Gaussian ensembles repulsion at small values is found. The ordered spacing statistics and their ratio are numerically studied for the integrable circle billiard, the chaotic cardioid billiard, the standard map and the zeroes of the Riemann zeta function. Very good agreement with the predictions is found."

G. Chavez and A. Allawala, "Prime zeta function statistics and Riemann zero-difference repulsion" (preprint 02/2021)

[abstract:] "We present a derivation of the numerical phenomenon that differences between the Riemann zeta function's nontrivial zeros tend to avoid being equal to the imaginary parts of the zeros themselves, a property called statistical ``repulsion'' between the zeros and their differences. Our derivation relies on the statistical properties of the prime zeta function, whose singularity structure specifies the positions of the Riemann zeros. We show that the prime zeta function on the critical line is asymptotically normally distributed with a covariance function that is closely approximated by the logarithm of the Riemann zeta function's magnitude on the $1$-line. This creates notable negative covariance at separations approximately equal to the imaginary parts of the Riemann zeros. This covariance function and the singularity structure of the prime zeta function combine to create a conditional statistical bias at the locations of the Riemann zeros that predicts the zero-difference repulsion effect. Our method readily generalizes to describe similar effects in the zeros of related Dirichlet $L$-functions."

G. Mussardo and A. LeClair, "Randomness of Mobius coefficents and brownian motion: Growth of the Mertens function and the Riemann Hypothesis" (preprint 01/2021)

[abstract:] "The validity of the Riemann Hypothesis (RH) on the location of the non-trivial zeros of the Riemann zeta-function is directly related to the growth of the Mertens function: the RH is indeed true if the Mertens function goes asymptotically as $M(x) \simeq x^{1/2+\epsilon}$. We show that this behavior can be established on the basis of a new probabilistic approach based on the global properties of the Mertens function. To this aim, we focus the attention on the square-free numbers and we derive a series of probabilistic results concerning the prime number distribution along the series of square-free numbers, the average number of prime divisors, the Erdos–Kac theorem for square-free numbers, etc. These results lead us to the conclusion that the Mertens function is subject to a normal distribution as much as any other random walk, therefore with an asymptotic behaviour given by $x^{1/2+\epsilon}$. We also argue how the Riemann Hypothesis implies the Generalised Riemann Hypothesis for the Dirichlet $L$-functions. Next we study the local properties of the Mertens function dictated by the Mobius coefficients restricted to the square-free numbers. We perform a massive statistical analysis on these coefficients, applying to them a series of randomness tests of increasing precision and complexity. The successful outputs of all these tests (with a level of confidence of 99% that all the sub-sequences analyzed are indeed random) can be seen as impressive experimental confirmations of the probabilistic normal law distribution of the Mertens function analytically established earlier. In view of the theoretical probabilistic argument and the large battery of statistical tests, we can conclude that while a violation of the RH is strictly speaking not impossible, it is however ridiculously improbable."

papers by A. Khrennikov, et al. on p-adic probability, etc.

A. Khrennikov , "p-adic probability and statistics", Dokl. Akad. Nauk 322 (1992) 1075-1079.

D. Dubischar, V.M. Gundlach, O. Steinkamp, and A. Khrennikov, "Attractors of random dynamical systems over p-adic numbers and a model of noisy cognitive processes", Physica D 130 (1999) 1-12

M. Endo and A. Khrennikov "Unboundedness of the p-adic Gaussian distribution" Izvestia Akad. Nauk USSR, ser. Matem., 56, no. 5, (1992) 1116-1103.

A. Khrennikov, "Axiomatics of the p-adic theory of probability", Dokl. Akad. Nauk 26, no. 5 (1992) 1075-1079.

A. Khrennikov, "p-adic statistical models", Dokl. Akad. Nauk 330, no. 3 (1993) 300-304.

A. Khrennikov, "p-adic probability theory and its applications. The principle of statistical stabization of frequencies", Theoretical and Mathematical Physics 97, no. 3 (1993) p. 348-363.

A. Khrennikov, "Bernoulli probabilities with p-adic values", Dokl. Akad. Nauk 338, No. 3 (1994) 313-316.

A. Khrennikov, "An algorithmic approach to p-adic probability theory", Dokl. Akad. Nauk 335, no. 1 (1994) 35-38.

A. Khrennikov, "On probablity distributions on the field of p-adic numbers", Theory of Probability and Applications 40 no.1 (1995) 189-192.

A. Khrennikov, "An extension of the frequency approach of R. von Mises and the axiomatic approach of N.A. Kolmogorov to the p-adic theory of probability", Theory of Probability and Applications 40, no. 2 (1995) 458-463.

A. Khrennikov, "The statistical simulation over fields of p-adic numbers", Matem. Modelirovanie 7, no. 4 (1995) 87-98.

A. Khrennikov, "A limit theorem for p-adic probabilities", Izvestia Akad. Nauk., ser. Matem., 59, no. 3 (1995) 207-223.

M. Endo and A. Khrennikov, "On the annihilators of the p-adic Gaussian distributions", Comm. Math. Univ. Sancti Pauli 144, no. 1 (1995) 105-108.

A. Khrennikov, "p-adic analogues of the law of large numbers and the central limit theorem", Indag. Math. 8 (1) (1997) 61-77.

A. Khrennikov, "p-adic asymptotic of Bernoulli probabilities", Theory of Probability and its Applications 42, no. 4 (1997) 839-845.

A. Khrennikov, "The Bernoulli theorem for probabilities that take p-adic values", Dokl. Akad. Nauk 354, no. 4 (1998) 461-464 [also Dokl. Math. 55 no. 3, 402-405.]

A. Khrennikov, "p-adic behaviour of Bernoulli probabilities", Statistics and Probability Letters 37, no. 4 (1998) 375-380.

A. Khrennikov, S. Yamada, and A. van Rooij, "Measure-theoretical approach to p-adic probability theory", Annals Math. Blaise Pascal 6, no. 1 (1999) 21-32.

A. Khrennikov, "p-adic information spaces, infinitely small probabilities and anomalous phenomena", Journal of Scientific Exploration 4, no. 13 (1999) 665-680.


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