×

Nicolaas Govert de Bruijn, the enchanter of friable integers. (English) Zbl 1295.11102

Summary: N. G. de Bruijn carried out fundamental work on integers having only small prime factors and the Dickman-de Bruijn function that arises on computing the density of those integers. In this he used his earlier work on linear functionals and differential-difference equations. We review his main contributions and also some later improvements by others authors.

MSC:

11N25 Distribution of integers with specified multiplicative constraints
01A70 Biographies, obituaries, personalia, bibliographies

Biographic References:

de Bruijn, Nicolaas Govert
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Alladi, K., The Turán-Kubilius inequality for integers without large prime factors, J. Reine Angew. Math., 335, 180-196 (1982) · Zbl 0483.10050
[2] Alladi, K., Asymptotic estimates of sums involving the Moebius function, J. Number Theory, 14, 86-98 (1982) · Zbl 0486.10004
[3] Alladi, K., Asymptotic estimates of sums involving the Moebius function. II, Trans. Amer. Math. Soc., 272, 87-105 (1982) · Zbl 0499.10050
[4] Alladi, K., An Erdős-Kac theorem for integers without large prime factors, Acta Arith., 49, 81-105 (1987) · Zbl 0627.10030
[6] Ankeny, N. C.; Onishi, H., The general sieve, Acta Arith., 10, 31-62 (1964/1965) · Zbl 0127.27002
[7] Arratia, R.; Barbour, A. D.; Tavaré, S., Random combinatorial structures and prime factorizations, Notices Amer. Math. Soc., 44, 903-910 (1997) · Zbl 0915.60011
[8] Arratia, R.; Barbour, A. D.; Tavaré, S., (Logarithmic Combinatorial Structures: A Probabilistic Approach. Logarithmic Combinatorial Structures: A Probabilistic Approach, EMS Monographs in Mathematics (2003), European Mathematical Society (EMS): European Mathematical Society (EMS) Zürich) · Zbl 1040.60001
[10] Bender, E. A., Asymptotic methods in enumeration, SIAM Rev., 16, 485-515 (1974) · Zbl 0294.05002
[11] Berndt, B. C., Ramanujan reaches his hand from his grave to snatch your theorems from you, Asia Pac. Math. Newsl., 1, 2, 8-13 (2011)
[12] Buchstab, A. A., On those numbers in arithmetical progression all of whose prime factors are small in order of magnitude, Dokl. Akad. Nauk. SSSR, 67, 5-8 (1949), (in Russian)
[13] Buchstab, A. A., On an asymptotic estimate of the number of numbers of an arithmetic progression which are not divisible by ‘relatively’ small prime numbers, Sbornik N.S., 28, 70, 165-184 (1951), (in Russian) · Zbl 0042.04203
[14] Canfield, E. R., The asymptotic behavior of the Dickman-de Bruijn function, (Proceedings of The Thirteenth Southeastern Conference on Combinatorics, Graph Theory and Computing (Boca Raton, Fla., 1982). Proceedings of The Thirteenth Southeastern Conference on Combinatorics, Graph Theory and Computing (Boca Raton, Fla., 1982), Congr. Numer., vol. 35 (1982)), 139-148 · Zbl 0511.10005
[15] Canfield, E. R.; Erdős, P.; Pomerance, C., On a problem of Oppenheim concerning ‘factorisatio numerorum’, J. Number Theory, 17, 1-28 (1983) · Zbl 0513.10043
[16] Car, M., Théorèmes de densité dans \(F_q [x]\), Acta Arith., 48, 145-165 (1987) · Zbl 0575.12017
[17] Chandrasekharan, K., (Arithmetical Functions. Arithmetical Functions, Die Grundlehren der mathematischen Wissenschaften, Band 167 (1970), Springer-Verlag: Springer-Verlag New York-Berlin)
[18] Chowla, S. D.; Briggs, W. E., On the number of positive integers \(\leqslant x\) all of whose prime factors are \(\leqslant y\), Proc. Amer. Math. Soc., 6, 558-562 (1955) · Zbl 0066.03403
[19] Chowla, S. D.; Vijayaraghavan, T., On the largest prime divisors of numbers, J. Indian Math. Soc. (N.S.), 11, 31-37 (1947) · Zbl 0039.03702
[20] de la Bretèche, R.; Tenenbaum, G., Propriétés statistiques des entiers friables, Ramanujan J., 9, 139-202 (2005) · Zbl 1152.11040
[21] de la Bretèche, R.; Tenenbaum, G., Entiers friables: inégalité de Turán-Kubilius et applications, Invent. Math., 159, 531-588 (2005) · Zbl 1182.11045
[22] de la Bretèche, R.; Tenenbaum, G., On the friable Turán-Kubilius inequality, (Manstavicius, E.; etal., Anal. Probab. Methods Number Theory (2012), TEV: TEV Vilnius), 259-265 · Zbl 1348.11068
[23] de Bruijn, N. G., On Mahler’s partition problem, Indag. Math., 10, 210-220 (1948) · Zbl 0030.34502
[24] de Bruijn, N. G., The asymptotically periodic behavior of the solutions of some linear functional equations, Amer. J. Math., 71, 313-330 (1949) · Zbl 0033.27002
[25] de Bruijn, N. G., On some linear functional equations, Publ. Math. Debrecen, 1, 129-134 (1950) · Zbl 0036.19501
[26] de Bruijn, N. G., On the number of uncancelled elements in the sieve of Eratosthenes, Indag. Math., 12, 247-256 (1950) · Zbl 0037.03001
[27] de Bruijn, N. G., On some Volterra integral equations of which all solutions are convergent, Indag. Math., 12, 257-265 (1950) · Zbl 0038.26602
[28] de Bruijn, N. G., The asymptotic behaviour of a function occurring in the theory of primes, J. Indian Math. Soc. (N.S.), 15, 25-32 (1951) · Zbl 0043.06502
[29] de Bruijn, N. G., On the number of positive integers \(\leqslant x\) and free of prime factors \(> y\), Indag. Math., 13, 50-60 (1951) · Zbl 0042.04204
[30] de Bruijn, N. G., The difference-differential equation \(F^\prime(x) = e^{\alpha x + \beta} F(x - 1)\), I, II, Indag. Math., 15, 449-458 (1953), 459-464 · Zbl 0053.38703
[31] de Bruijn, N. G., On the number of integers \(\leqslant x\) whose prime factors divide \(n\), Illinois J. Math., 6, 137-141 (1962) · Zbl 0100.03703
[32] de Bruijn, N. G., On the number of positive integers \(\leqslant x\) and free prime factors \(> y\). II, Indag. Math., 28, 239-247 (1966) · Zbl 0139.27203
[33] de Bruijn, N. G., Asymptotic Methods in Analysis (1981), Dover Publications, Inc.: Dover Publications, Inc. New York · Zbl 0556.41021
[34] de Bruijn, N. G., Jack van Lint (1932-2004), Nieuw Arch. Wiskd. (5), 6, 106-109 (2005), (in Dutch)
[36] de Bruijn, N. G.; van Lint, J. H., On the number of integers \(\leqslant x\) whose prime factors divide \(n\), Acta Arith., 8, 349-356 (1962/1963) · Zbl 0201.37601
[37] de Bruijn, N. G.; van Lint, J. H., Incomplete sums of multiplicative functions. I. II, Indag. Math., 26, 339-347 (1964), 348-359 · Zbl 0131.28703
[38] Dickman, K., On the frequency of numbers containing prime factors of a certain relative magnitude, Ark. Mat. Astr. Fys., 22, 1-14 (1930) · JFM 56.0178.04
[39] Ennola, V., On numbers with small prime divisors, Ann. Acad. Sci. Fenn. AI, 440, 16 pp (1969) · Zbl 0174.33903
[40] Erdős, P., Problem and solution no. 136, Wisk. Opgaven, 21, 133-135 (1963)
[41] Erdős, P.; Ivić, A.; Pomerance, C., On sums involving reciprocals of the largest prime factor of an integer, Glas. Mat. Ser. III, 21, 41, 283-300 (1986) · Zbl 0615.10055
[42] Evertse, J.-H.; Moree, P.; Stewart, C. L.; Tijdeman, R., Multivariate Diophantine equations with many solutions, Acta Arith., 107, 103-125 (2003) · Zbl 1026.11041
[43] Flajolet, P.; Sedgewick, R., Analytic Combinatorics (2009), Cambridge University Press: Cambridge University Press Cambridge · Zbl 1165.05001
[44] Friedlander, J.; Granville, A., Limitations to the equi-distribution of primes. I, Ann. of Math. (2), 129, 363-382 (1989) · Zbl 0671.10041
[45] Friedlander, J.; Granville, A.; Hildebrand, A.; Maier, H., Oscillation theorems for primes in arithmetic progressions and for sifting functions, J. Amer. Math. Soc., 4, 25-86 (1991) · Zbl 0724.11040
[46] Friedlander, J.; Iwaniec, H., (Opera de Cribro. Opera de Cribro, American Mathematical Society Colloquium Publications, vol. 57 (2010), AMS: AMS Providence, RI)
[47] Goldston, D. A.; McCurley, K. S., Sieving the positive integers by large primes, J. Number Theory, 28, 94-115 (1988) · Zbl 0639.10028
[49] Goncharov, W., Sur la distribution des cycles dans les permutations, C. R. (Dokl.) Acad. Sci. URSS, 35, 267-269 (1942) · Zbl 0063.01683
[50] Goncharov, W., On the field of combinatory analysis, Izv. Akad. Nauk SSSR. Izv. Akad. Nauk SSSR, Amer. Math. Soc. Transl., 19, 1-46 (1962), English translation in · Zbl 0129.31503
[51] Granville, A., Smooth numbers: computational number theory and beyond, (Algorithmic Number Theory: Lattices, Number Fields, Curves and Cryptography. Algorithmic Number Theory: Lattices, Number Fields, Curves and Cryptography, Math. Sci. Res. Inst. Publ., vol. 44 (2008), Cambridge Univ. Press: Cambridge Univ. Press Cambridge), 267-323 · Zbl 1230.11157
[52] Halberstam, H.; Richert, H.-E., (Sieve Methods. Sieve Methods, London Mathematical Society Monographs, vol. 4 (1974), Academic Press: Academic Press London-New York) · Zbl 0298.10026
[53] Hanrot, G.; Martin, B.; Tenenbaum, G., Constantes de Turán-Kubilius friables: une étude numérique, Exp. Math., 19, 345-361 (2010) · Zbl 1276.11152
[54] Hanrot, G.; Tenenbaum, G.; Wu, J., Moyennes de certaines fonctions multiplicatives sur les entiers friables. II, Proc. Lond. Math. Soc. (3), 96, 107-135 (2008) · Zbl 1195.11129
[55] Hensley, D., The number of positive integers \(\leqslant x\) and free of prime factors \(> y\), J. Number Theory, 21, 286-298 (1985) · Zbl 0575.10037
[56] Hensley, D., The convolution powers of the Dickman function, J. Lond. Math. Soc., 33, 395-406 (1986) · Zbl 0565.10041
[57] Hildebrand, A., Integers free of large prime factors and the Riemann hypothesis, Mathematika, 31, 258-271 (1984) · Zbl 0544.10042
[58] Hildebrand, A., On the number of positive integers \(\leqslant x\) and free of prime factors \(> y\), J. Number Theory, 22, 289-307 (1986) · Zbl 0575.10038
[59] Hildebrand, A., The asymptotic behavior of the solutions of a class of differential-difference equations, J. Lond. Math. Soc., 42, 11-31 (1990) · Zbl 0675.34037
[60] Hildebrand, A.; Tenenbaum, G., On integers free of large prime factors, Trans. Amer. Math. Soc., 296, 265-290 (1986) · Zbl 0601.10028
[61] Hildebrand, A.; Tenenbaum, G., Integers without large prime factors, J. Théor. Nombres Bordeaux, 5, 411-484 (1993) · Zbl 0797.11070
[62] Hildebrand, A.; Tenenbaum, G., On a class of differential-difference equations arising in number theory, J. Anal. Math., 61, 145-179 (1993) · Zbl 0797.11072
[63] Hua, L.-K., Estimation of an integral, Sci. Sin., 4, 393-402 (1951)
[64] Iwaniec, H., Rosser’s sieve, Acta Arith., 36, 171-202 (1980) · Zbl 0435.10029
[65] Knuth, D. E.; Trabb Pardo, L., Analysis of a simple factorization algorithm, Theoret. Comput. Sci., 3, 321-348 (1976/77) · Zbl 0362.10006
[66] Mahler, K., On a special functional equation, J. Lond. Math. Soc., 15, 115-123 (1940) · JFM 66.1214.04
[68] Maier, H., Primes in short intervals, Michigan Math. J., 32, 221-225 (1985) · Zbl 0569.10023
[69] Marsaglia, G.; Zaman, A.; Marsaglia, J. C., Numerical solution of some classical differential-difference equations, Math. Comp., 53, 191-201 (1989) · Zbl 0675.65073
[70] Martin, B.; Tenenbaum, G., Sur l’inégalité de Turán-Kubilius friable, J. Reine Angew. Math., 647, 175-234 (2010) · Zbl 1214.11109
[71] Mitchell, W. C., An evaluation of Golomb’s constant, Math. Comp., 22, 411-415 (1968)
[72] Montgomery, H. L.; Vaughan, R. C., (Multiplicative Number Theory. I. Classical Theory. Multiplicative Number Theory. I. Classical Theory, Cambridge Studies in Advanced Mathematics, vol. 97 (2007), Cambridge University Press: Cambridge University Press Cambridge) · Zbl 1142.11001
[74] Moree, P., A special value of Dickman’s function, Math. Student, 64, 47-50 (1995) · Zbl 1194.11106
[75] Moree, P., A generalization of the Buchstab equation, Manuscripta Math., 94, 267-270 (1997) · Zbl 0942.11042
[76] Moree, P., De zeeftheorie, Nieuw Arch. Wiskd. (5), 14, 1, 24-29 (2013), (in Dutch)
[78] Norton, K. K., (Numbers with Small Prime Factors, and The Least \(k\) th Power Non-residue. Numbers with Small Prime Factors, and The Least \(k\) th Power Non-residue, Memoirs of the American Mathematical Society, vol. 106 (1971), American Mathematical Society: American Mathematical Society Providence, R.I) · Zbl 0211.37801
[79] Odlyzko, A. M., Asymptotic enumeration methods, (Handbook of Combinatorics, Vols. 1, 2 (1995), Elsevier: Elsevier Amsterdam), 1063-1229 · Zbl 0845.05005
[80] Pennington, W. B., On Mahler’s partition problem, Ann. of Math. (2), 57, 531-546 (1953) · Zbl 0050.04005
[81] Ramanujan, S., The Lost Notebook and Other Unpublished Papers (1988), Narosa: Narosa New Delhi · Zbl 0639.01023
[82] Rankin, R. A., The difference between consecutive prime numbers, J. Lond. Math. Soc., 13, 242-247 (1938) · Zbl 0019.39403
[83] Rawsthorne, D. A., Selberg’s sieve estimate with a one sided hypothesis, Acta Arith., 41, 281-289 (1982) · Zbl 0436.10021
[84] Saias, É., Sur le nombre des entiers sans grand facteur premier, J. Number Theory, 32, 78-99 (1989) · Zbl 0676.10028
[85] Schwarz, W., Einige Anwendungen Tauberscher Sätze in der Zahlentheorie. B, J. Reine Angew. Math., 219, 157-179 (1965) · Zbl 0147.02404
[86] Scourfield, E. J., On some sums involving the largest prime divisor of \(n\). II, Acta Arith., 98, 313-343 (2001) · Zbl 0993.11047
[87] Selberg, A., On the normal density of primes in small intervals and the difference between consecutive primes, Arch. Math. Naturvid., 47, 87-105 (1943) · Zbl 0063.06869
[88] Shepp, L. A.; Lloyd, S. L., Ordered cycle lengths in a random permutation, Trans. Amer. Math. Soc., 121, 340-357 (1966) · Zbl 0156.18705
[89] Smida, H., Sur les puissances de convolution de la fonction de Dickman, Acta Arith., 59, 123-143 (1991) · Zbl 0881.11069
[90] Soundararajan, K., An asymptotic expression related to the Dickman function, Ramanujan J., 29, 25-30 (2012) · Zbl 1268.11138
[91] Specht, W., Zahlenfolgen mit endlich vielen Primteilern, S.-B. Math.-Nat. Kl. Bayer. Akad. Wiss., 1948, 149-169 (1949) · Zbl 0034.31503
[92] Tenenbaum, G., Lois de répartition des diviseurs. II, Acta Arith., 38, 1-36 (1980/81) · Zbl 0437.10028
[93] Tenenbaum, G., La méthode du col en théorie analytique des nombres, (Séminaire de Théorie des Nombres. Séminaire de Théorie des Nombres, Paris 1986-87. Séminaire de Théorie des Nombres. Séminaire de Théorie des Nombres, Paris 1986-87, Progr. Math., vol. 75 (1988), Birkhäuser Boston: Birkhäuser Boston Boston, MA), 411-441
[94] Tenenbaum, G., (Introduction to Analytic and Probabilistic Number Theory. Introduction to Analytic and Probabilistic Number Theory, Cambridge Studies in Advanced Mathematics, vol. 46 (1995), Cambridge University Press: Cambridge University Press Cambridge) · Zbl 0880.11001
[95] Tenenbaum, G., Crible d’Ératosthène et modèle de Kubilius, (Number Theory in Progress, Vol. 2 (1999), de Gruyter: de Gruyter Berlin), 1099-1129, Zakopane-Kościelisko, 1997 · Zbl 0936.11052
[96] Tenenbaum, G.; Wu, J., Moyennes de certaines fonctions multiplicatives sur les entiers friables, J. Reine Angew. Math., 564, 119-166 (2003) · Zbl 1195.11132
[97] Tenenbaum, G.; Wu, J., (Moyennes de Certaines Fonctions Multiplicatives sur les Entiers Friables. IV, Anatomy of Integers. Moyennes de Certaines Fonctions Multiplicatives sur les Entiers Friables. IV, Anatomy of Integers, CRM Proc. Lecture Notes, vol. 46 (2008), Amer. Math. Soc: Amer. Math. Soc Providence, RI), 129-141 · Zbl 1182.11042
[98] Toole, B. A., Ada, The Enchantress of Numbers: Prophet of the Computer Age (1998), Strawberry Press
[99] van Asch, B.; Blokhuis, A.; Hollmann, H.; Kantor, W.; van Tilborg, H., Jack van Lint (1932-2004): a survey of his scientific work, J. Combin. Theory Ser. A, 113, 1594-1613 (2006) · Zbl 1112.01017
[100] van de Lune, J.; Wattel, E., On the numerical solution of a differential-difference equation arising in analytic number theory, Math. Comp., 23, 417-421 (1969) · Zbl 0176.46602
[101] Wheeler, F. S., Two differential-difference equations arising in number theory, Trans. Amer. Math. Soc., 318, 491-523 (1990) · Zbl 0697.10035
[102] Xuan, T. Z., On the asymptotic behavior of the Dickman-de Bruijn function, Math. Ann., 297, 519-533 (1993) · Zbl 0786.11058
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.