R. Warlimont, Factorisatio numerorum with constraints, J. Number Theory 45 (1993) 186-199; MR 94f:11098.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....properties of the number and sum of distinct parts using an elementary approach. Another paper by Richmond and Knopfmacher [44] is also interesting since the results there further reveal the intricacy of the composition structure when studied from a distinct viewpoint. See also Warlimont [51] for a multiplicative counterpart. A closely related stochastic model to integer composition is the one studied by Chen [9] There he considered the number of distinct values assumed by a (finite) sequence of discrete random variables with total sum n. Although limit theorems for the number of ....

R. Warlimont. Factorisatio numerorum with constraints. Journal of Number Theory 45 (1993) 186--199.

.... 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 Problems column of The American Mathematical Monthly, volume 104, number 10 (1997) Notation. Throughout this paper, x is the major asymptotic parameter which is taken to be suciently large. The generic symbols , c, and K always represent suitably ....

R. Warlimont, Factorisatio numerorum with constraints, Journal of Number Theory 45 (1993), 186-199. 24

....properties of the number and sum of distinct parts using an elementary approach. Another paper by Richmond and Knopfmacher [44] is also interesting since the results there further reveal the intricacy of the composition structure when studied from a distinct viewpoint. See also Warlimont [51] for a multiplicative counterpart. A closely related stochastic model to integer composition is the one studied by Chen [9] There he considered the number of distinct values assumed by a (finite) sequence of discrete random variables with total sum n. Although limit theorems for the number of ....

R. Warlimont. Factorisatio numerorum with constraints. Journal of Number Theory 45 (1993) 186--199.

....dlog ke 3 log n log log n 4 bits (by Lemma 1, k 6 log n) Actually, u) can be represented in a more compact way. Indeed, the coding of cpath(u) is not the best possible. To improve the coding we need the following lemma originally proved by Kalm ar in 1930, and that can be also founded in [13]. Lemma 4 [13] Let Z(n) be the number of ordered integer sequences p 1 ; p 2 ; such that p i 2 and Q i 1 p i 6 n. Then, Z(n) n = with = where is the unique real solution of the equation ( 2 and is the Riemann Zeta function. We have 1:7286 and ....

....n log log n 4 bits (by Lemma 1, k 6 log n) Actually, u) can be represented in a more compact way. Indeed, the coding of cpath(u) is not the best possible. To improve the coding we need the following lemma originally proved by Kalm ar in 1930, and that can be also founded in [13] Lemma 4 [13] Let Z(n) be the number of ordered integer sequences p 1 ; p 2 ; such that p i 2 and Q i 1 p i 6 n. Then, Z(n) n = with = where is the unique real solution of the equation ( 2 and is the Riemann Zeta function. We have 1:7286 and 3:1429. From this ....

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R. Warlimont, Factorisatio numerorum with constraints, Journal of Number Theory, 45 (1993), pp. 186-199.

....) exp ae Gamma 1 X r=1 ( Gamma1) r i(rs)t r r oe = X ff ( Gammat) jffj c Gamma1 ff Y ff j 0 i(ff j s) i.e. i(fsg k ) Gamma1) k X jffj=k c Gamma1 ff Y ff j 0 i(ff j s) 4. 6) We note the following connection with factorisatio numerorum [51] See also [29, 70, 78]. Let ff be as in Definition 4.3. Define the unrestricted divisor function associated with the partition ff by d ff (m) X Q j1 d ff j j =m 1: MULTIPLE POLYLOGARITHMS: A BRIEF SURVEY 11 For example d 1;1 is the ordinary divisor function, and d 2 (m) 1 if m is a perfect square and zero ....

Richard Warlimont, Factorisatio numerorum with constraints, J. Number Theory, 45 (1993), 186--199.

....dlog ke 3 log n log log n 4 bits (by Lemma 1, k 6 log n) Actually, u) can be represented in a more compact way. Indeed, the coding of cpath(u) is not the best possible. To improve the coding we need the following lemma originally proved by Kalm ar in 1930, and that can be also founded in [14]. Lemma 4. 14] Let Z(n) be the number of ordered integer sequences p 1 ; p 2 ; such that p i 2 and Q i 1 p i 6 n. Then, Z(n) n = with = 0 ( where is the unique real solution of the equation ( 2 and is the Riemann Zeta function. We have 1:7286 and ....

....n log log n 4 bits (by Lemma 1, k 6 log n) Actually, u) can be represented in a more compact way. Indeed, the coding of cpath(u) is not the best possible. To improve the coding we need the following lemma originally proved by Kalm ar in 1930, and that can be also founded in [14] Lemma 4. [14] Let Z(n) be the number of ordered integer sequences p 1 ; p 2 ; such that p i 2 and Q i 1 p i 6 n. Then, Z(n) n = with = 0 ( where is the unique real solution of the equation ( 2 and is the Riemann Zeta function. We have 1:7286 and 3:1429. From ....

[Article contains additional citation context not shown here]

R. Warlimont, Factorisatio numerorum with constraints, Journal of Number Theory, 45 (1993), pp. 186-199.

....r i(rs)t r r oe = X ff ( Gammat) jffj c Gamma1 ff Y ff j 0 i(ff j s) 10 DOUGLAS BOWMAN AND DAVID M. BRADLEY i.e. i(fsg k ) Gamma1) k X jffj=k c Gamma1 ff Y ff j 0 i(ff j s) 4. 6) We note the following connection with factorisatio numerorum [47] See also [26, 68, 76]. Let ff be as in Definition 4.2. Define the unrestricted divisor function associated with the partition ff by d ff (m) X Q j1 d ff j j =m 1: For example d 1;1 is the ordinary divisor function, and d 2 (m) 1 if m is a perfect square and zero otherwise. Proposition 4.3. Let k (m) ....

Richard Warlimont, Factorisatio numerorum with constraints, J. Number Theory, 45 (1993), 186--199.

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R. Warlimont, Factorisatio numerorum with constraints, J. Number Theory 45 (1993) 186-199; MR 94f:11098.

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R. Warlimont, Factorisatio numerorum with constraints, Journal of Number Theory 45 (1993), 186--199. 30

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