On ideals free of large prime factors

Eira J. Scourfield[1]

  • [1] Mathematics Department, Royal Holloway University of London, Egham, Surrey TW20 0EX, UK.

Journal de Théorie des Nombres de Bordeaux (2004)

  • Volume: 16, Issue: 3, page 733-772
  • ISSN: 1246-7405

Abstract

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In 1989, E. Saias established an asymptotic formula for Ψ ( x , y ) = n ≤ x : p ∣ n ⇒ p ≤ y with a very good error term, valid for exp ( log log x ) ( 5 / 3 ) + ϵ ≤ y ≤ x , x ≥ x 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.

How to cite

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Scourfield, Eira J.. "On ideals free of large prime factors." Journal de Théorie des Nombres de Bordeaux 16.3 (2004): 733-772. <http://eudml.org/doc/249280>.

@article{Scourfield2004,
abstract = {In 1989, E. Saias established an asymptotic formula for $\Psi (x,y)=\left| \left\lbrace n\le x:p\mid n\Rightarrow p\le y\right\rbrace \right| $ with a very good error term, valid for $\exp \left( (\log \log x)^\{(5/3)+\epsilon \}\right) \le y\le x$, $x\ge x_\{0\}(\epsilon )$, $\epsilon &gt;0.$ We extend this result to an algebraic number field $K$ by obtaining an asymptotic formula for the analogous function $\Psi _\{K\}(x,y)$ with the same error term and valid in the same region. Our main objective is to compare the formulae for $\Psi (x,y)$ and $\Psi _\{K\}(x,y),$ and in particular to compare the second term in the two expansions.},
affiliation = {Mathematics Department, Royal Holloway University of London, Egham, Surrey TW20 0EX, UK.},
author = {Scourfield, Eira J.},
journal = {Journal de Théorie des Nombres de Bordeaux},
keywords = {smooth ideals; Dickman function; Dedekind zeta function},
language = {eng},
number = {3},
pages = {733-772},
publisher = {Université Bordeaux 1},
title = {On ideals free of large prime factors},
url = {http://eudml.org/doc/249280},
volume = {16},
year = {2004},
}

TY - JOUR
AU - Scourfield, Eira J.
TI - On ideals free of large prime factors
JO - Journal de Théorie des Nombres de Bordeaux
PY - 2004
PB - Université Bordeaux 1
VL - 16
IS - 3
SP - 733
EP - 772
AB - In 1989, E. Saias established an asymptotic formula for $\Psi (x,y)=\left| \left\lbrace n\le x:p\mid n\Rightarrow p\le y\right\rbrace \right| $ with a very good error term, valid for $\exp \left( (\log \log x)^{(5/3)+\epsilon }\right) \le y\le x$, $x\ge x_{0}(\epsilon )$, $\epsilon &gt;0.$ We extend this result to an algebraic number field $K$ by obtaining an asymptotic formula for the analogous function $\Psi _{K}(x,y)$ with the same error term and valid in the same region. Our main objective is to compare the formulae for $\Psi (x,y)$ and $\Psi _{K}(x,y),$ and in particular to compare the second term in the two expansions.
LA - eng
KW - smooth ideals; Dickman function; Dedekind zeta function
UR - http://eudml.org/doc/249280
ER -

References

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  1. N.G. de Bruijn, On the number of positive integers ≤ x and free of prime factors &gt; y . Indag. Math. 13 (1951), 50–60. Zbl0042.04204MR46375
  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. Zbl0043.06502MR43838
  3. J.A. Buchmann, C.S Hollinger, On smooth ideals in number fields. J. Number Theory 59 (1996), 82–87. Zbl0859.11060MR1399699
  4. K. Dilcher, Generalized Euler constants for arithmetical progressions. Math. Comp. 59 (1992), 259–282. Zbl0752.11033MR1134726
  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. Zbl0615.10055MR896810
  6. J.H. Evertse, P. Moree, C.L. Stewart, R. Tijdeman, Multivariate Diophantine equations with many solutions. Acta Arith. 107 (2003), 103–125. Zbl1026.11041MR1970818
  7. J.B. Friedlander, On the large number of ideals free from large prime divisors. J. Reine Angew. Math. 255 (1972), 1–7. Zbl0243.10034MR299579
  8. J.R. Gillett, On the largest prime divisors of ideals in fields of degree n. Duke Math. J. 37 (1970), 589–600. Zbl0211.07804MR268153
  9. L.J. Goldstein, Analytic Number Theory, Prentice-Hall, Inc., New Jersey, 1971. Zbl0226.12001MR498335
  10. D.G. Hazlewood,On ideals having only small prime factors. Rocky Mountain J. Math. 7 (1977), 753–768. Zbl0373.12004MR444591
  11. A. Hildebrand, On the number of positive integers ≤ x and free of prime factors &gt; y . J. Number Theory 22 (1986), 289–307. Zbl0575.10038MR831874
  12. A. Hildebrand, G. Tenenbaum, On integers free of large prime factors. Trans. Amer. Math. Soc. 296 (1986), 265–290. Zbl0601.10028MR837811
  13. M.N. Huxley, N. Watt, The number of ideals in a quadratic field II. Israel J. Math. 120 (2000), part A, 125–153. Zbl0977.11049MR1815373
  14. A. Ivić, Sum of reciprocals of the largest prime factor of an integer. Arch. Math. 36 (1981), 57–61. Zbl0436.10019MR612237
  15. A. Ivić, On some estimates involving the number of prime divisors of an integer. Acta Arith. 49 (1987), 21–33. Zbl0573.10038MR913761
  16. A. Ivić, On sums involving reciprocals of the largest prime factor of an integer II. Ibid 71 (1995), 229–251. Zbl0820.11052MR1339128
  17. A. Ivić, The Riemann zeta-function, Wiley, New York - Chichester - Brisbane - Toronto - Singapore, 1985. Zbl0556.10026MR792089
  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. Zbl0546.10037MR781162
  19. U. Krause, Abschätzungen für die Funktion Ψ K ( x , y ) in algebraischen Zahlkörpern. Manuscripta Math. 69 (1990), 319–331. Zbl0759.11028MR1078363
  20. E. Landau, Einführung in die elementare und analytische Theorie der algebraische Zahlen und Ideale. Teubner, Leipzig, 1927, reprint Chelsea, New York, 1949. Zbl0045.32202
  21. S. Lang, Algebraic Number Theory, Addison-Wesley, Reading, Mass. - Menlo Park, Calif. - London - Don Mills, Ont., 1970. Zbl0211.38404MR282947
  22. P. Moree, An interval result for the number field Ψ ( x , y ) function. Manuscripta Math. 76 (1992), 437–450. Zbl0772.11044MR1185030
  23. P. Moree, On some claims in Ramanujan’s ‘unpublished’ manuscript on the partition and tau functions, arXiv:math.NT/0201265. Zbl1066.11059MR2111687
  24. W. Narkiewicz, Elementary and analytic theory of algebraic numbers. PNW, Warsaw, 1974. Zbl0276.12002MR347767
  25. W.G. Nowak, On the distribution of integer ideals in algebraic number fields. Math. Nachr. 161 (1993), 59–74. Zbl0803.11061MR1251010
  26. E. Saias, Sur le nombre des entiers sans grand facteur premier. J. Number Theory 32 (1989), 78–99. Zbl0676.10028MR1002116
  27. E.J. Scourfield, On some sums involving the largest prime divisor of n, II. Acta Arith. 98 (2001), 313–343. Zbl0993.11047MR1829776
  28. H. Smida, Valeur moyenne des fonctions de Piltz sur entiers sans grand facteur premier. Ibid 63 (1993), 21–50. Zbl0769.11034MR1201617
  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. Zbl0153.07003MR223332
  30. W. Staś, On the order of Dedekind zeta-function in the critical strip. Funct. Approximatio Comment. Math. 4 (1976), 19–26. Zbl0357.12012MR439770
  31. G. Tenenbaum, Introduction to analytic and probabilistic number theory, CUP, Cambridge, 1995. Zbl0831.11001MR1366197

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