×

Quelques exemples de suites unimodales en théorie des nombres. (Some examples of unimodal sequences in number theory). (French) Zbl 0711.11031

A sequence \((u_ k)_{k\in I}\) of nonnegative real numbers defined over some interval I of integers is called unimodal if there exists \(k_ 0\in Z\cup \{\pm \infty \}\) such that \(u_ k\leq u_{k+1}\) for \(k<k_ 0\) and \(u_ k\geq u_{k+1}\) for \(k\geq k_ 0\). The paper under review gives a very readable survey of some unimodal sequences arising in various parts of mathematics. These sequences include partition functions such as the number p(n,k) of partitions of n into k parts; the sequence of coefficients of the polynomial \(\prod^{n}_{\ell =1}(1+x^{\ell});\) and arithmetic counting functions such as the number \(\pi\) (x,k) of positive integers \(n\leq x\) having exactly k distinct prime factors. The author describes in some detail an argument of A. M. Odlyzko and L. B. Richmond [Eur. J. Comb. 3, 69-84 (1982; Zbl 0482.10015)] proving the unimodality of the second example, and his own recent proof of the unimodality of \(\pi\) (x,k) [Ann. Inst. Fourier 40, 255-270 (1990; cf. the preceding review)]. He concludes with a list of unsolved problems.
Reviewer: A.Hildebrand

MSC:

11N25 Distribution of integers with specified multiplicative constraints
11P82 Analytic theory of partitions
11N37 Asymptotic results on arithmetic functions
PDFBibTeX XMLCite
Full Text: DOI Numdam EuDML

References:

[1] Almkvist, G., Proof of a conjecture about unimodal polynomials, J. of Number Theory32 (1989), 43-57. · Zbl 0678.05002
[2] Andrews, G.E., The theory of partitions, Addison-Wesley (1976), Reading. · Zbl 0371.10001
[3] Balazard, M., Delange, H. et Nicolas, J.-L., Sur le nombre de facteurs premiers des entiers, C.R.A.S.306 série I (1988), 511-514. · Zbl 0644.10032
[4] Balazard, M., Comportement statistique du nombre de facteurs premiers des entiers, Séminaire de Th. des Nombres, Paris 1987-1988, (1990), 1-21, Birkhaüser. · Zbl 0689.10047
[5] Balazard, M., Unimodalité de la distribution du nombre des diviseurs premiers d’un entier. A paraître aux Annales de l’Institut Fourier. · Zbl 0711.11030
[6] Comtet, L., Analyse combinatoire (1970), P.U.F, Paris. · Zbl 0221.05002
[7] Dharmadikari, S. et Joag-Dev, K., Unimodality, convexity and applications (1988), Academic Press, New-York. · Zbl 0646.62008
[8] Erdös, P., On the integers having exactly k prime factors, Annals of Math.49 (1948), 53-66. · Zbl 0030.29604
[9] Erdös, P. et Tenenbaum, G., Sur les densités de certaines suites d’entiers, Proc. London Math. Soc. (3)59 (1989), 417-438. · Zbl 0694.10040
[10] Gerber, H. et Keilson, J., Some results for discrete unimodality, J. Amer. Statist. Assoc.66 (1971), 386-389. · Zbl 0236.60017
[11] Hensley, D., The distribution of round numbers, Proc. London Math. Soc. (3) 54 (1987), 412-444. · Zbl 0588.10047
[12] Hildebrand, A. et Tenenbaum, G., On the number of prime factors of an integer, Duke Math. J.56 (1988), 471-501. · Zbl 0655.10036
[13] Hughes, J.W., Lie algebraic proofs of some theorems on partitions, Number Theory and Algebra (H. Zassenhaus, ed.) (1977), Academic Press, New-York. · Zbl 0372.10010
[14] Nicolas, J.-L., Sur la distribution des entiers ayant une quantité fixée de facteurs premiers, Acta Arith.44 (1984), 191-200. · Zbl 0512.10034
[15] Odlyzko, A.M. et Richmond, L.B., On the compositions of an integer, Combinatorial Mathematics VII. Proceedings (1979), 199-210, (R.W. Robinson et al. ed.). . · Zbl 0451.05009
[16] Odlyzko, A.M. et Richmond, L.B., On the unimodality of some partition polynomials, Europ. J. Combinatorics2 (1982), 69-84. · Zbl 0482.10015
[17] Odlyzko, A.M. et Richmond, L.B., On the unimodality of high powers of discrete distributions, Annals of Probability13 (1985), 299-306. · Zbl 0561.60021
[18] Pomerance, C., On the distribution of round numbers, Number Theory (K. Alladi ed.), (Proc. Ootacamund, India, 1984), 173-200, . · Zbl 0565.10038
[19] Roth, K.F. et Szekeres, G., Some asymptotic formulae in the theory of partitions, Quart. J. Math. Oxford (2) 5 (1954), 241-259. · Zbl 0057.03902
[20] Selberg, A., Note on a paper by L.G. Sathe, J. Indian Math. Soc.18 (1954), 83-87. · Zbl 0057.28502
[21] Stanley, R.P., Unimodal sequences arising from Lie algebras, Alfred Young Day Proceedings (T.V. Narayona, R.M. Mathsen and J.G. Williams eds.) (1980), Marcel Dekker, New-York. · Zbl 0451.05004
[22] Stanley, R.P., Log-concave and unimodal sequences in algebra, combinatories and geometry, Prépublication. · Zbl 0792.05008
[23] Szekeres, G., Some asymptotic formulae in the theory of partitions (II), Quart. J. Math. Oxford (2) 4 (1953), 96-111. · Zbl 0050.04101
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.