On ideals free of large prime factors
Journal de théorie des nombres de Bordeaux, Tome 16 (2004) no. 3, pp. 733-772.

En 1989, E. Saias a établi une formule asymptotique pour Ψ(x,y)=nx:pnpy avec un très bon terme d’erreur, valable si exp(loglogx) (5/3)+ϵ yx, xx 0 (ϵ), ϵ>0. Nous étendons ce résultat à un corps de nombre K en obtenant une formule asymptotique pour la fonction analogue Ψ K (x,y) avec le même terme d’erreur et la même zone de validité. Notre objectif principal est de comparer les formules pour Ψ(x,y) et Ψ K (x,y), en particulier comparer le second terme des développements.

In 1989, E. Saias established an asymptotic formula for Ψ(x,y)=nx:pnpy with a very good error term, valid for exp(loglogx) (5/3)+ϵ yx, xx 0 (ϵ), ϵ>0. We extend this result to an algebraic number field K by obtaining an asymptotic formula for the analogous function Ψ K (x,y) with the same error term and valid in the same region. Our main objective is to compare the formulae for Ψ(x,y) and Ψ K (x,y), and in particular to compare the second term in the two expansions.

DOI : 10.5802/jtnb.468
Scourfield, Eira J. 1

1 Mathematics Department, Royal Holloway University of London, Egham, Surrey TW20 0EX, UK.
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Scourfield, Eira J. On ideals free of large prime factors. Journal de théorie des nombres de Bordeaux, Tome 16 (2004) no. 3, pp. 733-772. doi : 10.5802/jtnb.468. http://www.numdam.org/articles/10.5802/jtnb.468/

[1] N.G. de Bruijn, On the number of positive integers x and free of prime factors >y. Indag. Math. 13 (1951), 50–60. | MR | Zbl

[2] N.G. de Bruijn, The asymptotic behaviour of a function occurring in the theory of primes.J. Indian Math. Soc. (NS) 15 (1951), 25–32. | MR | Zbl

[3] J.A. Buchmann, C.S Hollinger, On smooth ideals in number fields. J. Number Theory 59 (1996), 82–87. | MR | Zbl

[4] K. Dilcher, Generalized Euler constants for arithmetical progressions. Math. Comp. 59 (1992), 259–282. | MR | Zbl

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

[6] J.H. Evertse, P. Moree, C.L. Stewart, R. Tijdeman, Multivariate Diophantine equations with many solutions. Acta Arith. 107 (2003), 103–125. | MR | Zbl

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

[8] J.R. Gillett, On the largest prime divisors of ideals in fields of degree n. Duke Math. J. 37 (1970), 589–600. | MR | Zbl

[9] L.J. Goldstein, Analytic Number Theory, Prentice-Hall, Inc., New Jersey, 1971. | MR | Zbl

[10] D.G. Hazlewood,On ideals having only small prime factors. Rocky Mountain J. Math. 7 (1977), 753–768. | MR | Zbl

[11] A. Hildebrand, On the number of positive integers x and free of prime factors >y. J. Number Theory 22 (1986), 289–307. | MR | Zbl

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

[13] M.N. Huxley, N. Watt, The number of ideals in a quadratic field II. Israel J. Math. 120 (2000), part A, 125–153. | MR | Zbl

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

[15] A. Ivić, On some estimates involving the number of prime divisors of an integer. Acta Arith. 49 (1987), 21–33. | MR | Zbl

[16] A. Ivić, On sums involving reciprocals of the largest prime factor of an integer II. Ibid 71 (1995), 229–251. | MR | Zbl

[17] A. Ivić, The Riemann zeta-function, Wiley, New York - Chichester - Brisbane - Toronto - Singapore, 1985. | MR | Zbl

[18] A. Ivić, C. Pomerance, Estimates for certain sums involving the largest prime factor of an integer. In: Topics in Classical Number Theory (Budapest,1981), Colloq. Math. Soc. János Bolyai 34, North-Holland, Amsterdam, 1984, 769–789. | MR | Zbl

[19] U. Krause, Abschätzungen für die Funktion Ψ K (x,y) in algebraischen Zahlkörpern. Manuscripta Math. 69 (1990), 319–331. | MR | Zbl

[20] E. Landau, Einführung in die elementare und analytische Theorie der algebraische Zahlen und Ideale. Teubner, Leipzig, 1927, reprint Chelsea, New York, 1949. | Zbl

[21] S. Lang, Algebraic Number Theory, Addison-Wesley, Reading, Mass. - Menlo Park, Calif. - London - Don Mills, Ont., 1970. | MR | Zbl

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

[23] P. Moree, On some claims in Ramanujan’s ‘unpublished’ manuscript on the partition and tau functions, arXiv:math.NT/0201265. | MR

[24] W. Narkiewicz, Elementary and analytic theory of algebraic numbers. PNW, Warsaw, 1974. | MR | Zbl

[25] W.G. Nowak, On the distribution of integer ideals in algebraic number fields. Math. Nachr. 161 (1993), 59–74. | MR | Zbl

[26] E. Saias, Sur le nombre des entiers sans grand facteur premier. J. Number Theory 32 (1989), 78–99. | MR | Zbl

[27] E.J. Scourfield, On some sums involving the largest prime divisor of n, II. Acta Arith. 98 (2001), 313–343. | MR | Zbl

[28] H. Smida, Valeur moyenne des fonctions de Piltz sur entiers sans grand facteur premier. Ibid 63 (1993), 21–50. | MR | Zbl

[29] A.V. Sokolovskii, A theorem on the zeros of Dedekind’s zeta-function and the distance between ‘neighbouring’ prime ideals, (Russian). Ibid 13 (1968), 321–334. | MR | Zbl

[30] W. Staś, On the order of Dedekind zeta-function in the critical strip. Funct. Approximatio Comment. Math. 4 (1976), 19–26. | MR | Zbl

[31] G. Tenenbaum, Introduction to analytic and probabilistic number theory, CUP, Cambridge, 1995. | MR | Zbl

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