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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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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.
Benford’s law, values of Lfunctions and the 3x + 1 problem
 ACTA ARITH
, 2008
"... We show the leading digits of a variety of systems satisfying certain conditions follow Benford’s Law. For each system proving this involves two main ingredients. One is a structure theorem of the limiting distribution, specific to the system. The other is a general technique of applying Poisson Sum ..."
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We show the leading digits of a variety of systems satisfying certain conditions follow Benford’s Law. For each system proving this involves two main ingredients. One is a structure theorem of the limiting distribution, specific to the system. The other is a general technique of applying Poisson Summation to the limiting distribution. We show the distribution of values of Lfunctions near the central line and (in some sense) the iterates of the 3x+1 Problem are Benford.
Bernstein polynomials and Brownian motion
 Amer. Math. Monthly
"... One of the greatest pleasures in mathematics is the surprising connections that often appear between apparently disconnected ideas and theories. Some particularly striking instances exist in the interaction between probability theory and analysis. One of the simplest is the elegant proof of the ..."
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Cited by 4 (1 self)
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One of the greatest pleasures in mathematics is the surprising connections that often appear between apparently disconnected ideas and theories. Some particularly striking instances exist in the interaction between probability theory and analysis. One of the simplest is the elegant proof of the
RamanujanFourier series, the WienerKhintchine formula and the distribution of prime pairs
, 1999
"... The WienerKhintchine formula plays a central role in statistical mechanics. It is shown here that the problem of prime pairs is related to autocorrelation and hence to a WienerKhintchine formula. "Experimental" evidence is given for this. c # 1999 Elsevier Science B.V. All rights res ..."
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Cited by 4 (2 self)
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The WienerKhintchine formula plays a central role in statistical mechanics. It is shown here that the problem of prime pairs is related to autocorrelation and hence to a WienerKhintchine formula. "Experimental" evidence is given for this. c # 1999 Elsevier Science B.V. All rights reserved. PACS: 05.40+j; 02.30.Nw; 02.10.Lh Keywords: Twin primes; RamanujanFourier series; WienerKhintchine formula 1. Introduction " The WienerKhintchine theorem states a relationship between two important characteristics of a random process: the power spectrum of the process and the correlation function of the process" [1]. One of the outstanding problems in number theory is the problem of prime pairs which asks how primes of the form p and p+h (where h is an even integer) are distributed. One immediately notes that this is a problem of #nding correlation between primes. We make two key observations. First of all there is an arithmetical function (a function de#ned on integers) which traps the...
Sieving and the ErdősKac Theorem
, 2006
"... We give a relatively easy proof of the ErdősKac theorem via computing moments. We show how this proof extends naturally in a sieve theory context, and how it leads to several related results in the literature. ..."
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We give a relatively easy proof of the ErdősKac theorem via computing moments. We show how this proof extends naturally in a sieve theory context, and how it leads to several related results in the literature.
On the Twin and Cousin Primes
, 1996
"... The computer results of the investigation of the number of pairs of primes separated by gap d = 2 ("twins") and gap d = 4 ("cousins") are reported. The number of twins and cousins turn out to be is almost the same. The plot of the function W (x) giving the difference of the numbe ..."
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The computer results of the investigation of the number of pairs of primes separated by gap d = 2 ("twins") and gap d = 4 ("cousins") are reported. The number of twins and cousins turn out to be is almost the same. The plot of the function W (x) giving the difference of the number of twins and cousins for x 2 (1; 10 12 ) is presented . This function has fractal properties and the fractal dimension is approximately 1.48  what is very close to the fractal dimension of the usual Brownian motion. The set of primes, up to which the numbers of twins and cousins are exactly the same seems to have the fractal structure. The boxcounting method gives the fractal dimension of this set approximately 0.51. The statistics of distances between primes being the zeros of W (x) display the crossover from the exponential decrease to the power like dependence with the exponent equal 1.48. Arguments that W (x) has the same properties as a typical sample path of the random walk are given. The analog of the Brun's constant is defined for cousins. 2 Marek Wolf 1. In the paper [1] Hardy and Littlewood have proposed about 15 conjectures. The conjecture B of their paper states 1 : There are infinitely many primes pairs p; p 0 = p + d; (1) for every even d. If d (x) is the number of pairs less than x, then d (x) c 2 x ln 2 (x) Y pjd p \Gamma 1 p \Gamma 2 : (2) Here the constant c 2 (sometimes called "twinprime " constant, see [3]) is defined in the following way: c 2 j 2 Y p?2 ` 1 \Gamma 1 (p \Gamma 1) 2 ' = 1:32032 . . . (3) Nobody has proved as yet (2), even there is no proof that there is infinity of twin (d = 2) primes. The largest twins known officially are: 697053813 \Theta 2 16352 \Sigma 1 (4) found recently by Indlekofer and Jarai, [2]. The pairs of ...
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"... Abstract. We give a relatively easy proof of the ErdősKac theorem via computing moments. We show how this proof extends naturally in a sieve theory context, and how it leads to several related results in the literature. Let ω(n) denote the number of distinct prime factors of the natural number n. ..."
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Abstract. We give a relatively easy proof of the ErdősKac theorem via computing moments. We show how this proof extends naturally in a sieve theory context, and how it leads to several related results in the literature. Let ω(n) denote the number of distinct prime factors of the natural number n. The average value of ω(n) as n ranges over the integers below x is