Approximating the number of integers without large prime factors
HTML articles powered by AMS MathViewer
- by Koji Suzuki PDF
- Math. Comp. 75 (2006), 1015-1024 Request permission
Abstract:
$\Psi (x,y)$ denotes the number of positive integers $\leq x$ and free of prime factors $>y$. Hildebrand and Tenenbaum gave a smooth approximation formula for $\Psi (x,y)$ in the range $(\log x)^{1+\epsilon }< y \leq x$, where $\epsilon$ is a fixed positive number $\leq 1/2$. In this paper, by modifying their approximation formula, we provide a fast algorithm to approximate $\Psi (x,y)$. The computational complexity of this algorithm is $O(\sqrt {(\log x)(\log y)})$. We give numerical results which show that this algorithm provides accurate estimates for $\Psi (x,y)$ and is faster than conventional methods such as algorithms exploiting Dickman’s function.References
- A. O. L. Atkin and D. J. Bernstein, Prime sieves using binary quadratic forms, Math. Comp. 73 (2004), no. 246, 1023–1030. MR 2031423, DOI 10.1090/S0025-5718-03-01501-1
- Eric Bach and René Peralta, Asymptotic semismoothness probabilities, Math. Comp. 65 (1996), no. 216, 1701–1715. MR 1370848, DOI 10.1090/S0025-5718-96-00775-2
- Daniel J. Bernstein, Bounding smooth integers (extended abstract), Algorithmic number theory (Portland, OR, 1998) Lecture Notes in Comput. Sci., vol. 1423, Springer, Berlin, 1998, pp. 128–130. MR 1726065, DOI 10.1007/BFb0054856
- E. R. Canfield, Paul Erdős, and Carl Pomerance, On a problem of Oppenheim concerning “factorisatio numerorum”, J. Number Theory 17 (1983), no. 1, 1–28. MR 712964, DOI 10.1016/0022-314X(83)90002-1
- A. Y. Cheer and D. A. Goldston, A differential delay equation arising from the sieve of Eratosthenes, Math. Comp. 55 (1990), no. 191, 129–141. MR 1023043, DOI 10.1090/S0025-5718-1990-1023043-8
- N. G. de Bruijn, On the number of positive integers $\leq x$ and free of prime factors $>y$, Nederl. Acad. Wetensch. Proc. Ser. A. 54 (1951), 50–60. MR 0046375
- N. G. de Bruijn, The asymptotic behaviour of a function occurring in the theory of primes, J. Indian Math. Soc. (N.S.) 15 (1951), 25–32. MR 43838
- K. Dickman, On the frequency of numbers containing prime factors of a certain relative magnitude, Arkiv För Matematik, Astromi Fysik. Band 22 A, no. 10, 1-14, 1930.
- Adolf Hildebrand, On the number of positive integers $\leq x$ and free of prime factors $>y$, J. Number Theory 22 (1986), no. 3, 289–307. MR 831874, DOI 10.1016/0022-314X(86)90013-2
- Adolf Hildebrand, On the local behavior of $\Psi (x,y)$, Trans. Amer. Math. Soc. 297 (1986), no. 2, 729–751. MR 854096, DOI 10.1090/S0002-9947-1986-0854096-0
- Adolf Hildebrand and Gérald Tenenbaum, On integers free of large prime factors, Trans. Amer. Math. Soc. 296 (1986), no. 1, 265–290. MR 837811, DOI 10.1090/S0002-9947-1986-0837811-1
- Adolf Hildebrand and Gérald Tenenbaum, Integers without large prime factors, J. Théor. Nombres Bordeaux 5 (1993), no. 2, 411–484. MR 1265913, DOI 10.5802/jtnb.101
- Simon Hunter and Jonathan Sorenson, Approximating the number of integers free of large prime factors, Math. Comp. 66 (1997), no. 220, 1729–1741. MR 1423076, DOI 10.1090/S0025-5718-97-00874-0
- George Marsaglia, Arif Zaman, and John C. W. Marsaglia, Numerical solution of some classical differential-difference equations, Math. Comp. 53 (1989), no. 187, 191–201. MR 969490, DOI 10.1090/S0025-5718-1989-0969490-3
- Karl K. Norton, Numbers with small prime factors, and the least $k$th power non-residue, Memoirs of the American Mathematical Society, No. 106, American Mathematical Society, Providence, R.I., 1971. MR 0286739
- R. A. Rankin, The difference between consecutive prime numbers, J. London Math. Soc. 13, 242-247, 1938.
- Jonathan P. Sorenson, A fast algorithm for approximately counting smooth numbers, Algorithmic number theory (Leiden, 2000) Lecture Notes in Comput. Sci., vol. 1838, Springer, Berlin, 2000, pp. 539–549. MR 1850632, DOI 10.1007/10722028_{3}6
- Koji Suzuki, An estimate for the number of integers without large prime factors, Math. Comp. 73 (2004), no. 246, 1013–1022. MR 2031422, DOI 10.1090/S0025-5718-03-01571-0
Additional Information
- Koji Suzuki
- Affiliation: Corporate Research Group, Fuji Xerox, 430, Sakai, Nakai-machi, Ashigarakami-gun, Kanagawa 259-0157, Japan
- Email: kohji.suzuki@fujixerox.co.jp
- Received by editor(s): September 30, 2004
- Received by editor(s) in revised form: December 13, 2004
- Published electronically: December 2, 2005
- © Copyright 2005 American Mathematical Society
- Journal: Math. Comp. 75 (2006), 1015-1024
- MSC (2000): Primary 11N25; Secondary 11Y05
- DOI: https://doi.org/10.1090/S0025-5718-05-01798-9
- MathSciNet review: 2199567