M. Balazard, H. Delange, and J.-L. Nicolas, Sur le nombre de facteurs premiers des entiers, Comptes Rendus de l'Academie des Sciences, Serie I, Paris, 306:511--514 (1988). 15  Home/Search   Document Not in Database   Summary   Related Articles   Check

....behaviours of #m (#) have been explicitly characterized, we can employ #m (#) as primitive asymptotic approximant for more sophisticated problems. Besides the classical Poisson approximations (cf. 5] let us mention the distribution of integers x with a given number of prime factors (cf. [4, 44]) and the number of components in decomposable combinatorial structures (cf. 21] Thus we investigate the asymptotic behaviour of #m (#) as # and m runs through its possible values (depending on #) When # is bounded and the asymptotic behaviour of #m (#) can be easily derived by the usual ....

M. Balazard, H. Delange, and J.-L. Nicolas, Sur le nombre de facteurs premiers des entiers, Comptes Rendus de l'Academie des Sciences, Serie I, Paris, 306:511--514 (1988). 15

....and negative binomial distributions is then naturally introduced. Comparing these results with those of the quantity ## x, m) ## x, m) 1#n#x # n) m 1, x 1, m N) ## n) being the number (multiplicities counted) of prime factors of n, we can, following Balazard, Delange and Nicolas [5], make the Gaussian transition of N between the Poisson and geometric behaviours more explicit by introducing more primitive approximants. Theorem 6 Let 0 B # 2 and set X = # #) log(n m) j , h(w) g(w #) w) 1 # #)w ##(1 #w #) # = max min B, 1) X , ....

M. Balazard, H. Delange, and J.-L. Nicolas. Sur le nombre de facteurs premiers des entiers, Comptes Rendus de l'Academie des Sciences, Serie I, Paris, 306 (1988), 511--514.

....are, however, much more involved. Finally, distributional properties associated with Oppenheim s problem of factorisatio numerorum (the unordered counterpart of Kalm ar s problem) having bivariate generating function n;m z 1 zn can be dealt with using the methods developed in [1, 11] and asymptotic properties of the modi ed Bessel functions; cf. 8, 12, 18, 20, 22] In particular, we have a Bessel geometric convolution law for the quantity n x n;m (properly normalized) For related materials, see [3, 6, 7, 10, 19, 17, 23] and page 969 (fourth paragraph) of the Unsolved ....

M. Balazard, H. Delange, and J.-L. Nicolas, Sur le nombre de facteurs premiers des entiers, Comptes Rendus de l'Academie des Sciences, Serie I, Paris 306 (1988), 511-514.

....36g ## (1) x 72# P(Y = m) 36g # (1) 72# for m = # x # #, x = o(# ) Thus 2# x A. The result (6) follows as above. The basic idea of the proof is the same as that in Balazard, Delange and Nicolas [11] (cf. also [10] for details) in which a uniform asymptotic formula was derived for the number of prime factors of a randomly chosen integer between 1 and n. The details having been given in [10] and in [56] for similar integrals, we shall only sketch the main steps. Set #m (#) P(Y m h) e ....

Balazard, M., Delange, H., and Nicolas, J.-L. (1988) Sur le nombre de facteurs premiers des entiers. Comptes Rendus de l'Academie des Sciences, Serie I, Paris 306, 511--514.

....behaviours of Pi m ( have been explicitly characterized, we can employ Pi m ( as primitive asymptotic approximant for more sophisticated problems. Besides the classical Poisson approximations (cf. 5] let us mention the distribution of integers x with a given number of prime factors (cf. [4, 44]) and the number of components in decomposable combinatorial structures (cf. 21] Thus we shall investigate the asymptotic behaviour of Pi m ( as 1 and m runs through its possible values (depending on ) When is bounded and m 1, the asymptotic behaviour of Pi m ( can be easily derived by ....

M. Balazard, H. Delange, and J.-L. Nicolas, Sur le nombre de facteurs premiers des entiers, Comptes Rendus de l'Acad'emie des Sciences, S'erie I, Paris, 306:511--514 (1988). 18

....of such structures can be found in [1] Moreover, most of our formulae subsist even in the case when q is a positive real number 1. 1 1) The proof of this theorem is, besides some peculiarities of di#erent inversion formulae, more involved than the one by Balazard, Delange, and Nicolas [5] for the number of integers x with m prime factors. In their context, the integral in question has a simple pole and a saddle point; while in our case, a pole of order q and a saddle point is encountered in the integrand; cf. also [25] The theorem is useful only if the asymptotic behaviors of ....

M. Balazard, H. Delange, and J.-L. Nicolas, Sur le nombre de facteurs premiers des entiers, Comptes Rendus de l'Academie des Sciences, Serie I, Paris, 306, 511--514 (1988).

....and negative binomial distributions is then naturally introduced. Comparing these results with those of the quantity x; m) x; m) 1 n x n) m 1; x 1; m 2 N) n) being the number (multiplicities counted) of prime factors of n, we can, following Balazard, Delange and Nicolas [5], make the Gaussian transition of N between the Poisson and geometric behaviours more explicit by introducing more primitive approximants. Theorem 6 Let 0 B and set X = log(n m) k (X) j , h(w) g(w= 1 w) 1 = w (1 w= max f1; min fB; ....

M. Balazard, H. Delange, and J.-L. Nicolas. Sur le nombre de facteurs premiers des entiers, Comptes Rendus de l'Academie des Sciences, Serie I, Paris, 306 (1988), 511-514.

....most of our formulae subsist even in the case when q is a positive real number 1. K being a sufficiently large number. r min ; 1 (m 1) The proof of this theorem is, besides some peculiarities of different inversion formulae, more involved than the one by Balazard, Delange, and Nicolas [5] for the number of integers x with m prime factors. In their context, the integral in question has a simple pole and a saddle point; while in our case, a pole of order q and a saddle point is encountered in the integrand; cf. also [25] The theorem is useful only if the asymptotic behaviors of ....

M. Balazard, H. Delange, and J.-L. Nicolas, Sur le nombre de facteurs premiers des entiers, Comptes Rendus de l'Acad'emie des Sciences, S'erie I, Paris, 306, 511--514 (1988).

....are, however, much more involved. Finally, distributional properties associated with Oppenheim s problem of factorisatio numerorum (the unordered counterpart of Kalmar s problem) having bivariate generating function m#1# n,m z zn s can be dealt with using the methods developed in [1, 13] and asymptotic properties of the modified Bessel functions; cf. 10, 14, 20, 23, 25] In particular, we have a Bessel geometric convolution law for the quantity n,m (properly normalized) For related materials, see [3] and page 969 (fourth paragraph) of the Unsolved Problems column of The ....

M. Balazard, H. Delange, and J.-L. Nicolas, Sur le nombre de facteurs premiers des entiers, Comptes Rendus de l'Academie des Sciences, Serie I, Paris 306 (1988), 511--514.

....(1) x 72 P(Y = m) 72 for m = x ) Thus j 1=ff uniformly for jxj A. The result (6) follows as above. The basic idea of the proof is the same as that in Balazard, Delange and Nicolas [11] (cf. also [10] for details) in which a uniform asymptotic formula was derived for the number of prime factors of a randomly chosen integer between 1 and n. The details having been given in [10] and in [56] for similar integrals, we shall only sketch the main steps. Set Pi m ( P(Y m h) e ....

Balazard, M., Delange, H., and Nicolas, J.-L. (1988) Sur le nombre de facteurs premiers des entiers. Comptes Rendus de l'Acad'emie des Sciences, S'erie I, Paris 306, 511--514.

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC