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On the asymptotic distribution of large prime factors
 J. London Math. Soc
, 1993
"... A random integer N, drawn uniformly from the set {1,2,..., n), has a prime factorization of the form N = a1a2...aM where ax ^ a2>... ^ aM. We establish the asymptotic distribution, as «» • oo, of the vector A(«) = (loga,/logiV: i:> 1) in a transparent manner. By randomly reordering the comp ..."
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Cited by 20 (0 self)
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A random integer N, drawn uniformly from the set {1,2,..., n), has a prime factorization of the form N = a1a2...aM where ax ^ a2>... ^ aM. We establish the asymptotic distribution, as «» • oo, of the vector A(«) = (loga,/logiV: i:> 1) in a transparent manner. By randomly reordering the components of A(«), in a sizebiased manner, we obtain a new vector B(n) whose asymptotic distribution is the GEM distribution with parameter 1; this is a distribution on the infinitedimensional simplex of vectors (xv x2,...) having nonnegative components with unit sum. Using a standard continuity argument, this entails the weak convergence of A(/i) to the corresponding PoissonDirichlet distribution on this simplex; this result was obtained by Billingsley [3]. 1.
Asymptotics of Poisson approximation to random discrete distributions: an analytic approach
 Advances in Applied Probability
, 1998
"... this paper, we shall describe the asymptotic behaviors of several distances of Poisson approximation to a wide class of discrete distributions covering many examples from number theory, combinatorics and arithmetic semigroups. Our aim is to show that whenever (analytic) generating functions of the r ..."
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Cited by 16 (9 self)
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this paper, we shall describe the asymptotic behaviors of several distances of Poisson approximation to a wide class of discrete distributions covering many examples from number theory, combinatorics and arithmetic semigroups. Our aim is to show that whenever (analytic) generating functions of the random variables in question are available, complexanalytic methods can be used to derive precise asymptotic results for the five distances above. Actually, we shall consider the following generalized distances: let ff ? 0 be a fixed positive number, (X; Y ) = FM (X; Y ) = (X; Y ) = sup K (X; Y ) = sup M (X; Y ) = jP(X = j) \Gamma P(Y = j) Note that d TV = d M . Besides the case ff = 1 (and ff = 1=2 for d M ), only the case d TV was previously studied by Franken [39] for Poisson approximation to the sum of independent but not identically distributed Bernoulli random variables. We take these quantities as our measures of degree of nearness of Poisson approximation, some of which may be interpreted as certain norms in suitable space as many authors did (cf. [12, 22, 23, 74, 96]). For a large class of discrete distributions, we shall derive an asymptotic main term together with an error estimate for each of these distances. Our results are thus "approximation theorems" rather than "limit theorems". The common form of the underlying structure of these distributions suggests the study of an analytic scheme as we did previously for normal approximation and large deviations (cf. [53, 54]). Many concrete examples from probabilistic number theory and combinatorial structures will justify the study of this scheme. Our treatment being completely general, many extensions can be further pursued with essentially the same line of methods. We shall di...
Some InformationTheoretic Computations . . .
 SUBMITTED TO JORMA RISSANEN’S FESTSCHRIFT VOLUME
, 2008
"... We illustrate how elementary informationtheoretic ideas may be employed to provide proofs for wellknown, nontrivial results in number theory. Specifically, we give an elementary and fairly short proof of the following asymptotic result, p≤n log p p ∼ log n, as n → ∞, where the sum is over all prim ..."
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We illustrate how elementary informationtheoretic ideas may be employed to provide proofs for wellknown, nontrivial results in number theory. Specifically, we give an elementary and fairly short proof of the following asymptotic result, p≤n log p p ∼ log n, as n → ∞, where the sum is over all primes p not exceeding n. We also give finiten bounds refining the above limit. This result, originally proved by Chebyshev in 1852, is closely related to the celebrated prime number theorem.
Distribution of Multiplicative Functions Defined on Semigroups
"... The value distribution problem for realvalued multiplicative functions defined on an additive arithmetical semigroup is examined. We prove that, in contrast to the classical theory of numbertheoretic functions defined on the semigroup of natural numbers, this problem is equivalent to that for addi ..."
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The value distribution problem for realvalued multiplicative functions defined on an additive arithmetical semigroup is examined. We prove that, in contrast to the classical theory of numbertheoretic functions defined on the semigroup of natural numbers, this problem is equivalent to that for additive functions only under some extra condition. In this way, applying the known results for additive functions we derive general sufficient conditions for the existence of a limit law for appropriately normalized multiplicative functions.
Some InformationTheoretic Computations Related to the Distribution of Prime Numbers
, 2007
"... We illustrate how elementary informationtheoretic ideas may be employed to provide proofs for wellknown, nontrivial results in number theory. Specifically, we give an elementary and fairly short proof of the following asymptotic result, p≤n log p p ∼ log n, as n → ∞, where the sum is over all prim ..."
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We illustrate how elementary informationtheoretic ideas may be employed to provide proofs for wellknown, nontrivial results in number theory. Specifically, we give an elementary and fairly short proof of the following asymptotic result, p≤n log p p ∼ log n, as n → ∞, where the sum is over all primes p not exceeding n. We also give finiten bounds refining the above limit. This result, originally proved by Chebyshev in 1852, is closely related to the celebrated prime number theorem.
Toward a Mathematical Definition of "Life"
, 1979
"... In discussions of the nature of life, the terms "complexity," "organism, " and "information content," are sometimes used in ways remarkably analogous to the approach of algorithmic information theory, a mathematical discipline which studies the amount of information nec ..."
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In discussions of the nature of life, the terms "complexity," "organism, " and "information content," are sometimes used in ways remarkably analogous to the approach of algorithmic information theory, a mathematical discipline which studies the amount of information necessary for computations. We submit that this is not a coincidence and that it is useful in discussions of the nature of life to be able to refer to analogous precisely defined concepts whose properties can be rigorously studied. We propose and discuss a measure of degree of organization and structure of geometrical patterns which is based on the algorithmic version of Shannon's concept of mutual information. This paper is intended as a contribution to von Neumann's program of formulating mathematically the fundamental concepts of biology in a very general setting, i.e. in highly simplified model universes.