Integers without large prime factors
Journal de théorie des nombres de Bordeaux, Volume 5 (1993) no. 2, pp. 411-484.
@article{JTNB_1993__5_2_411_0,
     author = {Adolf Hildebrand and Gerald Tenenbaum},
     title = {Integers without large prime factors},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {411--484},
     publisher = {Universit\'e Bordeaux I},
     volume = {5},
     number = {2},
     year = {1993},
     doi = {10.5802/jtnb.101},
     zbl = {0797.11070},
     mrnumber = {1265913},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.101/}
}
TY  - JOUR
TI  - Integers without large prime factors
JO  - Journal de théorie des nombres de Bordeaux
PY  - 1993
DA  - 1993///
SP  - 411
EP  - 484
VL  - 5
IS  - 2
PB  - Université Bordeaux I
UR  - https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.101/
UR  - https://zbmath.org/?q=an%3A0797.11070
UR  - https://www.ams.org/mathscinet-getitem?mr=1265913
UR  - https://doi.org/10.5802/jtnb.101
DO  - 10.5802/jtnb.101
LA  - en
ID  - JTNB_1993__5_2_411_0
ER  - 
%0 Journal Article
%T Integers without large prime factors
%J Journal de théorie des nombres de Bordeaux
%D 1993
%P 411-484
%V 5
%N 2
%I Université Bordeaux I
%U https://doi.org/10.5802/jtnb.101
%R 10.5802/jtnb.101
%G en
%F JTNB_1993__5_2_411_0
Adolf Hildebrand; Gerald Tenenbaum. Integers without large prime factors. Journal de théorie des nombres de Bordeaux, Volume 5 (1993) no. 2, pp. 411-484. doi : 10.5802/jtnb.101. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.101/

M. Abramowitz & I. Stegun, (1964) Handbook of Mathematical Functions, National Bureau of Standards, Wash ington 1964; Tenth Printing, Dover, New York, 1972.

W.R. Alford, A. Granville, & C. Pomerance, (1993) There are infinitely many Carmichael numbers, preprint. | MR

K. Alladi, (1982a) Asymptotic estimates of sums involving the Moebius function. II, Trans. Amer. Math. Soc. 272, 87-105. | MR | Zbl

(1982b) The Turán-Kubilius inequality for integers without large prime factors, J. Reine Angew. Math. 335, 180-196. | MR | Zbl

(1987) An Erdös-Kac theorem for integers without large prime factors, Acta Arith. 49, 81-105. | MR | Zbl

K. Alladi and P. Erdös, (1977) On an additive arithmetic function, Pacific J. Math. 71, 275-294. | MR | Zbl

(1979) On the asymptotic behavior of large prime factors of integers, Pacific J. Math. 82, 295-315. | MR | Zbl

A. Balog, (1984) p + a without large prime factors, in: Séminaire de Théorie des Nombres, Bordeaux 1983-84, Univ. Bordeaux, Talence, Exp. No. 31, 5 pp. | MR | Zbl

(1987) On the distribution of integers having no large prime factor, in: Journées Arithmétiques, Besançon 1985, Astérisque 147/148, pp. 27-31. | MR | Zbl

(1989) On additive representations of integers, Acta Math. Hungar. 54, 297-301. | MR | Zbl

A. Balog, P. Erdös, & G. Tenenbaum, (1990) On arithmetic functions involving consecutive divisors, in: Analytic Number Theory (B. C. Berndt et al., eds.), Proc. Conf. in Honor of Paul T. Bateman, Birkhäuser, Progress in Math. 85, 77-90. | MR | Zbl

A. Balog & C. Pomerance, (1992) The distribution of smooth numbers in arithmetic progressions, Amer. Math. Soc. 115, 33-43. | MR | Zbl

A. Balog & A. Sárközy, (1984a) On sums of integers having small prime factors. I, Stud. Sci. Math. Hungar. 19, 35-47. | MR | Zbl

(1984b) On sums of integers having small prime factors. II, Stud. Sci. Math. Hungar. 19, 81-88. | Zbl

F. Beukers, (1975) The lattice-points of n-dimensional tedrahedra, Indag. Math. 37, 365-372. | MR | Zbl

P. Billingsley, (1972) On the distribution of large prime divisors, Period. Math. Hungar. 2, 283-289. | MR | Zbl

A.E. Brouwer, (1974) Two number theoretic sums, Math. Centrum Amsterdam, Report ZW 19/74. | MR | Zbl

N. G. De Bruijn, (1951a) The asymptotic behaviour of a function occurring in the theory of primes, J. Indian Math. Soc. (N.S.) 15, 25-32. | MR | Zbl

(1951b) On the number of positive integers ≤ x and free of prime factors > y, Nederl. Akad. Wetensch. Proc. Ser. A 54, 50-60. | Zbl

(1966) On the number of positive integers ≤ x and free of prime factors > y, II, Nederl. Akad. Wetensch. Proc. Ser. A 69, 239-247. | Zbl

N. G. De Bruijn & J. H. Van Lint, (1964) Incomplete sums of multiplicative functions I, II, Nederl. Akad. Wetensch. Proc. Ser. A 67, 339-347; 348-359. | MR | Zbl

A.A. Buchstab, (1949) On those numbers in an arithmetic progression all prime factors of which are small in magnitude (Russian), Dokl. Akad. Na u k SSSR 67, 5-8. | MR

E.R. Canfield, (1982) On the asymptotic behavior of the Dickman-de Bruijn function, Congr. Numer. 38, 139-148. | MR | Zbl

E R. Canfield, P. Erdös, & C. Pomerance, (1983) On a problem of Oppenheim concerning "Factorisatio Numerorum", J. Number Theory 17, 1-28. | MR | Zbl

M. Car, (1987) Théorèmes de densité dans Fq[X], Acta Arith. 48, 145-165. | MR | Zbl

S.D. Chowla and T. Vijayaraghavan, (1947) On the largest prime divisors of numbers, J. Indian Math. Soc. (N. S.) 11, 31-37. | MR | Zbl

J.-M. De Koninck & D. Hensley, (1978) Sums taken over n ≤ x with prime factors ≤ y of zΩ(n), and their derivatives with respect to z, J. Indian Math. Soc. (N. S.) 42, 353-365. | Zbl

J.-M. De Koninck & A. Ivić, (1984a) Sommes de réciproques de grandes fonctions additives, Publ. Inst. Math. (Beograd) (N. S.) 35, 41-48. | MR | Zbl

(1984b) The distribution of the average prime divisor of an integer, Arch. Math. 43, 37-43. | Zbl

J.-M. De Koninck & A. Mercier, (1989) Les fonctions arithmetiques et le plus grand facteur premier, Acta Arith. 52, 25-48. | MR | Zbl

J.-M. Deshouillers & H. Iwaniec, (1982) On the greatest prime factor of n2+1, Ann. Inst. Fourier (Grenoble) 32, 1-11. | Numdam | MR | Zbl

K. Dickman, (1930) On the frequency of numbers containing prime factors of a certain relative magnitude, Ark. Mat. Astr. Fys. 22, 1-14. | JFM

R. Eggleton & J.L. Selfridge, (1976) Consecutive integers with no large prime factors, J. Austral. Math. Soc. Ser. A 22, 1-11. | MR | Zbl

V. Ennola, (1969) On numbers with small prime divisors, Ann. Acad. Sci. Fenn. Ser. AI 440, 16 pp. | MR | Zbl

P. Erdös, (1935) On the normal number of prime factors of p -1 and some other related problems concerning Euler's Φ-function, Quart. J. Math. (Oxford) 6, 205-213. | JFM

(1952) On the greatest prime factor of Π f (k), J. London Math. Soc. 2 7, 379-384. | Zbl

(1955) On consecutive integers, Nieuw Arch. Wisk. (3) 3, 124-128. | MR | Zbl

P. Erdös & A. Ivić, (1980) Estimates for sums involving the largest prime factor of an integer and certain related additive functions, Stud. Sci. Math. Hungar. 15, 183-199. | MR | Zbl

P. Erdös, A. Ivić, and C. Pomerance, (1986) On sums involving reciprocals of the largest prime factor of an integer, Glas. Mat. III. Ser. 21 (41), 283-300. | MR | Zbl

P. Erdös & A. Schinzel, (1990) On the greatest prime factor of Πx k=1 f(k), Acta Arith. 55, 191-200. | Zbl

A.S. Fainleib, (1967) On the estimation from below of the number of numbers with small prime factors (Russian), Dokl. Akad. Nauk USSR 7, 3-5. | MR

E. Fouvry & F. Grupp, (1986) On the switching principle in sieve theory, J. Reine Angew. Math. 370, 101-126. | MR | Zbl

E. Fouvry & G. Tenenbaum, (1990) Diviseurs de Titchmarsh des entiers sans grand facteur premier, in: Analytic Number Theory (E. Fouvry, ed.), Tokyo 1988, Lecture Notes in Math. 1434, Springer, Berlin, pp. 86-102. | MR | Zbl

(1991) Entiers sans grand facteur premier en progressions arithmétiques, London Math. Soc. (3) 63, 449-494. | MR | Zbl

J.B. Friedlander, (1972) On the number of ideals free of large prime divisors, J. Reine Angew. Math. 255, 1-7. | MR | Zbl

(1973a) On the least kth power residue in an algebraic number field, London Math. Soc. (3) 26, 19-34. | Zbl

(1973b) Integers without large prime factors, Nederl. Akad. Wetensch. Proc. Ser. A 76, 443-451. | MR | Zbl

(1976) Integers free from large and small primes, Proc. London Math. Soc. (3) 33, 565-576. | MR | Zbl

(1981) Integers without large prime factors. II, Acta Arith. 39, 53-57. | MR | Zbl

(1984a) Large prime factors in small intervals, in: Topics in classical number theory, Colloqu. Math. Soc. Janos Bolyai 34, North-Holland, Amsterdam, 511-518. | MR | Zbl

(1984b) Integers without large prime factors. III, Arch. Math. 43, 32-36. | MR | Zbl

(1985) Notes on Chebyshev's method, Rocky Mountain J. Math. 15, 377-383. | MR | Zbl

(1989) Shifted primes without large prime factors, in: Number Theory and Applications (R. A. Mollin, ed.), Kluver, pp. 393-401. | MR

J.B. Friedlander & J.C. Lagarias, (1987) On the distribution in short intervals of integers having no large prime factor, J. Number Theory 25, 249-273. | MR | Zbl

J. Galambos, (1976) The sequence of prime divisors of integers, Acta Arith. 31, 213-218. | MR | Zbl

J.R. Gillet, (1970) On the largest prime divisor of ideals in fields of degree n, Duke Math. J. 37, 589-600. | MR | Zbl

D.A. Goldston & K.S. Mccurley, (1988) Sieving the positive integers by large primes, J. Number Theory 28, 94-115. | MR | Zbl

A. Granville, (1989) On positive integers ≤ x with prime factors ≤ tlog x, in: Number Theory and Applications (R. A. Mollin, ed.), Kluver, pp. 403-422. | Zbl

(1991a) The lattice points of an n-dimensional tedrahedron, Aequationes Math. 41, 234-241. | MR | Zbl

(1991b) On pairs of coprime integers with no large prime factors, Expos. Math. 9, 335-350. | MR | Zbl

(1993a) Integers, without large prime factors, in arithmetic progressions. I, Acta Math. 170, 255-273. | MR | Zbl

(1993b) Integers, without large prime factors, in arithmetic progressions. II, preprint.

J.L. Hafner, (1993) On smooth numbers in short intervals under the Riemann Hypothesis, preprint.

H. Halberstarn & H.-E. Richert, (1971) Mean value theorems for a class of arithmetic functions, Acta Arith. 18, 243-256; Add. & Corr., ibid. 28, 107-110. | MR | Zbl

(1974) Sieve Methods, Academic Press, London, New York, San Francisco. | MR

H. Halberstam & K.F. Roth, (1951) On the gaps between consecutive k-free integers, J. London Math. Soc. 26, 268-273. | MR | Zbl

G. Harman, (1991) Short intervals containing numbers without large prime factors, Math. Proc. Cambridge Philos. Soc. 109, 1-5. | MR | Zbl

D.G. Hazlewood, (1973) On integers all of whose prime factors are small, Bull. London Math. Soc. 5, 159-163. | MR | Zbl

(1975a) On k-free integers with small prime factors, Proc. Amer. Math. Soc. 52, 40-44. | MR | Zbl

(1975b) Sums over positive integers with few prime factors, J. Number Theory 7, 189-207. | MR | Zbl

(1975c) On sums over Gaussian integers, Trans. Amer. Math. Soc. 209, 295-310. | MR | Zbl

(1977) On ideals having only small prime factors, Rocky Mountain J. Math. 7, 753-768. | MR | Zbl

D.R. Heath-Brown, (1987) Consecutive almost-primes, J. Indian Math. Soc. (N. S.) 52, 39-49. | MR | Zbl

D. Hensley, (1985) The number of positive integers ≤ x and free of prime divisors > y, J. Number Theory 21, 286-298. | Zbl

(1986) A property of the counting function of integers with no large prime factors, J. Number Theory 22, 46-74. | MR | Zbl

(1987) The distribution of Ω(n) among numbers with no large prime factors, in: Analytic Number Theory and Diophantine Problems (A. Adolphson et al., eds.), Proc. of a Conf. at Oklahoma State University 1984, Birkhauser, Progress in Math. 70, 247-281. | Zbl

A. Hildebrand, (1984a) Integers free of large prime factors and the Riemann Hypothesis, Mathematika 31, 258-271. | MR | Zbl

(1984b) On a problem of Erdös and Alladi, Monatsh. Math. 97, 119-124. | MR | Zbl

(1985a) Integers free of large prime factors in short intervals, Quart. J. Math. (Oxford) (2) 36, 57-69. | Zbl

(1985b) On a conjecture of A. Balog, Proc. Amer. Math. Soc. 95, 517-523. | Zbl

(1986a) On the number of positive integers < x and free of prime factors > y, J. Number Theory 22, 289-307. | MR | Zbl

(1986b) On the local behavior of Ψ(x, y), Trans. Amer. Math. Soc. 297, 729-751. | Zbl

(1987) On the number of prime factors of integers without large prime divisors, J. Number Theory 25, 81-106. | MR | Zbl

(1989) Integer sets containing strings of consecutive integers, Mathematika 36, 60-70. | MR | Zbl

A. Hildebrand & G. Tenenbaum, (1986) On integers free of large prime factors, Trans. Amer. Math. Soc. 296, 265-290. | MR | Zbl

(1993) On a class of difference differential equations arising in number theory, J. d'Analyse Math. 61, 145-179. | MR | Zbl

N.A. Hmyrova, (1964) On polynomials with small prime divisors (Russian), Dokl. Akad. Nauk SSSR 155, 1268-1271. | MR | Zbl

(1966) On polynomials with small prime divisors. II (Russian), Izv. Akad. Nauk SSSR Ser. Mat. 30, 1367-1372. | MR

C. Hooley, (1967) On the greatest prime factor of a quadratic polynomial, Acta Math. 117, 2-16. | MR | Zbl

(1978) On the greatest prime factor of a cubic polynomial, J. Reine Angew. Math. 303/304, 21-50. | MR | Zbl

A. Ivić, (1981) Sum of reciprocals of the largest prime factor of an integer, Arch. Math. 36, 57-61. | MR | Zbl

(1985a) The Riemann zeta-function, Wiley, New York. | Zbl

(1985b) On squarefree numbers with restricted prime factors, Stud. Sci. Math. Hungar. 20, 183-187. | MR

(1987) On some estimates involving the number of prime divisors of an integer, Acta Arith. 49, 21-33. | MR | Zbl

B. Hornfeck, (1959) Zur Verteilung gewisser Primzahlprodukte, Math. Ann. 139, 14-30. | MR | Zbl

A. Ivić and C. Pomerance, (1984) Estimates for certain sums involving the largest prime factor of an integer, in: Topics in classical number theory, Colloq. Budapest 1981, Colloq. Math. Soc. Janos Bolyai 34, pp. 769-789. | MR | Zbl

A. Ivić and G. Tenenbaum, (1986) Local densities over integers free of large prime factors, Quart. J. Math. (Oxford) (2) 37, 401-417. | MR | Zbl

J.H. Jordan, (1965) The divisibility of Gaussian integers by large Gaussian primes, Duke Math. J. 32, 503-510. | MR | Zbl

M. Jutila, (1974) On numbers with a large prime factor. II, J. Indian Math. Soc.(N. S.) 38, 125-130. | MR | Zbl

D.E. Knuth & L. Trabb Pardo, (1976) Analysis of a simple factorization algorithm, Theoret. Comput. Sci. 3, 321-348. | MR | Zbl

U. Krause, (1990) Abschätzungen für die Funktion ΨK (x,y) in algebraischen Zahlkörpern, Manuscripta Math. 69, 319-331. | Zbl

M. Langevin, (1975) Plus grand facteur premier d'entiers voisins, C. R. Acad. Sci. Paris Sér. A-B 281, A491-A493. | MR | Zbl

D.H. Lehmer & E. Lehmer, (1941) On the first case of Fermat's last theorem, Bull. Amer. Math. Soc. 47, 139-142. | JFM | MR | Zbl

H.W. Lenstra (1987) Factoring integers with elliptic curves, Ann. Alath. 126, 649-673. | MR | Zbl

B.V. Levin & U. Chariev, (1986) Sums of multiplicative functions with respect to numbers with prime divisors from given intervals (Russian), Dokl. Akad. Nauk. Tadzh. SSSR 29, 383-387. | MR | Zbl

B.V. Levin & A.S. Fainleib, (1967) Application of some integral equations to problems in number theory, Russian Math. Surveys 22, 119-204. | MR | Zbl

J. H. Van Lint & H.-E. Richert, (1964) Über die Surmme Σn≤x,P(n)≤ y μ2(n)/ϕ(n), Nederl. Akad. Wetensch. Proc. Ser. A 67, 582-587. | Zbl

S.P. Lloyd, (1984) Ordered prime divisors of a random integer, Ann. Probability 12, 1205-1212. | MR | Zbl

R. Lovorn, (1992) Rigorous, subexponential algorithms for discrete logarithms over finite fields, Ph.D. thesis, Univ. of Georgia, 1992.

J. Van De Lune, (1974) The truncated average limit and one of its applications in analytic number theory, Math. Centrum Amsterdam, Report ZW 20/74. | MR | Zbl

H.L. Montgomery, (1971) Topics in Multiplicative Number Theory, Lecture Notes in Math. 227, Springer, Berlin. | MR | Zbl

P. Moree, (1992) An interval result for the number field Ψ (x, y ) function, Manuscripta Math. 76, 437-450. | Zbl

(1993) Psixyology and Diophantine equations, Thesis, Leiden University. | MR | Zbl

P. Moree & C.L. Stewart, (1990) Some Ramanujan-Nagell equations with many solutions, Indag. Math.(N. S.) 1, 465-472. | MR | Zbl

Y. Motohashi, (1976) An induction principle for the generalization of Bombieri's prime number theorem, Proc. Japan Acad. 52, 273-275. | MR | Zbl

(1979) A note on almost primes in short intervals, Proc. Japan Acad. Sci. Ser. A Math. Sci. 55 (1979), 225-226. | MR | Zbl

M. Naimi, (1988) Les entiers sans facteur carre ≤ x dont les facteurs premiers sont ≤y, in: Groupe de travail en théorie analytique des nombres 1986-87, Publ. Math. Orsay 88-01, pp. 69-76. | Zbl

T. Nagell, (1921) Generalisation d'un théorème de Tchébycheff, J. Math. Pur. Appl. (8) 4, 343-356. | JFM

K.K. Norton, (1968) Upper bounds for kth power coset representatives modulo n, Acta Arith. 15, 161-179. | MR | Zbl

(1971) Numbers with small prime factors and the least kth power non residue, Memoirs of the Amer. Math. Soc. 106. | Zbl

A.M. Odlyzko, (1985) Discrete logarithms in finite fields and their cryptographic significance, in Advances in Cryptology: Proceedings of Eurocrypt '84, (T. Beth, N. Cot, I. Ingemarsson, eds.), Lecture Notes in Computer Science 209, Springer-Verlag. | MR | Zbl

(1993) Discrete logarithms and smooth polynomials, preprint.

C. Pomerance, (1980) Popular values of Euler's function, Mathematika 27, 84-89. | MR | Zbl

(1987) Fast, rigorous factorization and discrete logarithm algorithms, in: Discrete Algorithms and Complexity (Kyoto, 1986), Academic Press, Boston. | MR | Zbl

K. Ramachandra, (1969) A note on numbers with a large prime factor. J. London Math. Soc. (2) 1, 303-306. | MR | Zbl

(1970) A note on numbers with a large prime factor. II, J. Indian Math. Soc.(N. S.) 34, 39-48. | MR | Zbl

(1971) A note on numbers with a large prime factor. III, Acta Arith. 19, 49-62. | MR | Zbl

K. Ramachandra & T.N. Shorey, (1973) On gaps between numbers with a large prime factor, Acta Arith. 24, 99-111. | MR | Zbl

V. Ramaswami, (1949) On the number of positive integers less than x and free of prime divisors greater than xc, Bull. Amer. Math. Soc. 55, 1122-1127. | MR | Zbl

(1951) Number of integers in an assigned arithmetic progression, ≤ x and prime to primes greater than xc, Proc. Amer. Math. Soc. 2, 318-319. | Zbl

R.A. Rankin, (1938) The difference between consecutive prime numbers, J. London Math. Soc. 13, 242-247. | JFM | Zbl

J. B. Van Rongen, (1975) On the largest prime divisor of an integer, Indag. Math. 37, 70-76. | MR | Zbl

E. Saias, (1989) Sur le nombre des entiers sans grand facteur premier, J. Number Theory 3 2, 78-99. | MR | Zbl

(1992) Entiers sans grand ni petit facteur premier. I, Acta Arith. 61, 347-374. | MR | Zbl

E.J. Scourfield, (1991) On some sums involving the largest prime divisor of n, Acta Arith. 59, no 4, 339-363. | MR | Zbl

T.N. Shorey, (1973) On gaps between numbers with a large prime factor II, Acta Arith. 25, 365-373. | MR | Zbl

H. Smida, (1991) Sur les puissances de convolution de la fonction de Dickman, Acta Arith. 59, no 2, 123-143. | MR | Zbl

(1993) Valeur moyenne des fonctions de Piltz sur les entiers sans grand facteur premier, Acta Arith. 63, no 1, 21-50. | MR | Zbl

W. Specht, (1949) Zahlenfolgen mit endlich vielen Primteilern, S.-B. Bayer. Akad. Wiss. Math. Nat. Abt. 1948, 149-169. | MR | Zbl

G. Tenenbaum, (1985) Sur les entiers sans grand facteur premier, in: Séminaire de Théorie des Nombres, Bordeaux 1984-85, Univ. Bordeaux I, Talence, 12 pp. | MR

(1988) La méthode du col en théorie analytique des nombres, in: Séminaire de Théorie des Nombres (C. Goldstein, ed.), Paris 1985-86, Birkhäuser, Progress in Math. 75, 411-441. | MR | Zbl

(1990a) Introduction à la théorie analytique et probabiliste des nombres, Publ. Inst. Elie Cartan, Vol. 13, Univ. Nancy 1. | Zbl

(1990b) Sur un problème d'Erdös et Alladi, in: Séminaire de Théorie des Nombres (C. Goldstein, ed.), Paris 1988-89, Birkhäuser, Progress in Math. 91, 221-239. | Zbl

(1990c) Sur une question d'Erdös et Schinzel II, Inventiones Math. 99, 215-224. | Zbl

(1993) Cribler les entiers sans grand facteur premier, in: R.C.Vaughan (ed.) Theory and applications of numbers without large prime factors, Phal. Trans. Royal l Soc. London, series A, to appear. | MR

R. Tijdeman, (1972) On the maximal distance of numbers with a large prime factor, J. London Math. Soc. (2) 5, 313-320. | MR | Zbl

(1973) On integers with many small prime factors, Compositio Math. 26, 319-330. | Numdam | MR | Zbl

(1974) On the maximal distance between integers composed of small primes, Compositio Math. 28, 159-162. | Numdam | MR | Zbl

N.M. Timofeev, (1977) Polynomials with small prime divisors (Russian), Taskent Gos. Univ. Naun. Trudy 548 Voprosy Mat., 87-91. | MR | Zbl

E.C. Titchmarsh and D.R. Heath-Brown, (1986) The theory of the Riemann zeta function, Second Edition, Oxford Univ. Press, Oxford. | MR | Zbl

J. Turk, (1980) Prime divisors of polynomials at consecutive integers, J. Reine Angew. Math. 319, 142-152. | MR | Zbl

(1982) Products of integers in short intervals, Report 8228M, Econometric Institut, Erasmus University, Rotterdam.

R.C. Vaughan, (1989) A new iterative method in Waring's problem, Acta Math. 162, 1-71. | MR | Zbl

A.M. Vershik, (1987) The asymptotic distribution of factorizations of natural numbers into prime divisors, Soviet Math. Dokl. 34, 57-61. | Zbl

I.M. Vinogradov, (1926) On a bound for the least nth power non-residue (Russian), Izv. Akad. Nauk SSSR 20, 47-58; English transl.: Trans. Amer. Math. Soc. 29 (1927), 218-226. | JFM

Y. Wang, (1964) Estimation and application of a character sum (Chinese), Shuzue Jinzhan 7, 78-83. | MR | Zbl

R. Warlimont, (1990) Sieving by large prime factors, Monatshefte Math. 109, 247-256. | MR | Zbl

(1991) Arithmetical semigroups, II: Sieving by large and small prime elements. Sets of multiples, Manuscripta Math. 71, 197-221. | MR | Zbl

F. Wheeler, (1990) Two differential-difference equations arising in number theory, Trans. Amer. Math. Soc. 318, 491-523. | MR | Zbl

D. Wolke, (1971) Polynomial values with small prime divisors, Acta Arith. 19, 327-333. | MR | Zbl

(1973) Über die mittlere Verteilung der Werte zahlentheoretischer Funktionen auf Restklassen. I, Math. Ann. 202, 1-205. | MR | Zbl

T. Wooley, (1992) Large improvements in Waring's problem, Annals Math.(2) 135, no 1, 135-164. | MR | Zbl

T.Z. Xuan, (1988) On a problem of Erdös and Ivić, Publ. Inst. Math. (Beograd) (N. S.) 43 (57), 9-15. | MR | Zbl

(1989a) On sums involving reciprocals of certain large additive functions. I, Publ. Inst. Math. (Beograd) (N. S.) 45 (59), 41-55. | MR | Zbl

(1989b) On sums involving reciprocals of certain large additive functions. II, Publ. Inst. Math. (Beograd) (N. S.) 46 (60), 25-32. | MR | Zbl

(1990) The average of dk(n) over integers free of large prime factors, Acta Arith. 55, 249-260. | MR | Zbl

(1991) The average of the divisor function over integers without large prime factors, Chinese Ann. of Math. Ser. A 12, 28-33. | MR | Zbl

(1993) On the asymptotic behavior of the DicZanan-de Bruijn function, Math. Ann. 297, 519-533. | MR | Zbl

Cited by Sources: