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Elements of the Random Walk: An Introduction for Advanced Students and Researche

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Rabin, "Probabilistic algorithm for testing primality", Journal of Number Theory 12 Shlesinger, "On the Riemann hypothesis: a fractal random walk approach", Physica A [abstract:] "In his investigation of the distribution of prime numbers Riemann, in , 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.

Elements Of The Random Walk An Introduction For Advanced Students And Researchers

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. Murty and M. Smith and P. 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. Dodson and J. 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. Lucio and Y. We specialize this general situation to three important families of distributions.

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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. Alexander, K. Baclawski and G.

Rota, "A stochastic interpretation of the Riemann zeta function" , Proceedings of the National Academy of Sciences USA 90 [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.

Albeverio and W. Joshi and S. Nagaraja, et al. Calude , P.

Elements of the Random Walk: An introduction for Advanced Students and Researchers

Hertling and B. Khoussainov, "Do the zeros of Riemann's zeta-function form a random sequence? Chang and Y. Peres , "Ladder Heights, Gaussian random walks, and the Riemann zeta function" , Annals of Probability , 25 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. Biane , J.

Pitman , and M. Pitman and M.

Yor, "Infinitely divisible laws associated with hyperbolic functions" , Canadian Journal of Mathematics 55 no. The distributions of C 1 and S 2 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 S t and C t by operations such as Brownian subordinations, exponential tilting, and weak limits, and characterized in various ways. Hughes, J. Keating, and N. Asmussen, P. Glynn and J. Bender, S. Boettcher, L. Mead, "Random walks in noninteger dimension" , Journal of Mathematical Physics 35 "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.

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This result has a number-theoretic interpretation. Karwowski, "A random walk on p -adics - the generator and its spectrum" , Stochastic Processes and their Applications , 53 S. Albeverio, W. Karwowski and X. Wolf , "Random walk on the prime numbers", Physica A Vardi , "Deterministic percolation" , Communications in Mathematical Physics [excerpt from introduction:] " 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. 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. Vardi, "Prime percolation" , Experimental Mathematics 7 [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 another token, I advance a probabilistic interpretation of Weil's positivity criterion, as opposed to the usual geometrical analogies or goals.

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. Donati-Martin, Zhan Shi and M. Castro and J. Mahecha, "Fractal supersymmetric quantum mechanics, geometric probability and the Riemann Hypothesis" , International Journal of Geometric Methods in Modern Physics 1 no.

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. The Geometric Probability to find a zero and an infinity of zeros in the imaginary axis is exactly unity.