Abstract
Let P(n) denote the largest prime divisor of n, and let Ψ(x,y) be the number of integers n≤x with P(n)≤y. In this paper we present improvements to Bernstein’s algorithm, which finds rigorous upper and lower bounds for Ψ(x,y). Bernstein’s original algorithm runs in time roughly linear in y. Our first, easy improvement runs in time roughly y 2/3. Then, assuming the Riemann Hypothesis, we show how to drastically improve this. In particular, if logy is a fractional power of logx, which is true in applications to factoring and cryptography, then our new algorithm has a running time that is polynomial in logy, and gives bounds as tight as, and often tighter than, Bernstein’s algorithm.
This work was supported by a grant from the Holcomb Research Institute. We wish to thank the referee, whose comments helped improve this paper.
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References
Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions. Dover (1970)
Atkin, A.O.L., Bernstein, D.J.: Prime sieves using binary quadratic forms. Mathematics of Computation 73, 1023–1030 (2004)
Bernstein, D.J.: Enumerating and counting smooth integers, ch. 2. PhD Thesis, University of California at Berkeley (May 1995)
Bernstein, D.J.: Bounding smooth integers. In: Buhler, J.P. (ed.) ANTS 1998. LNCS, vol. 1423, pp. 128–130. Springer, Heidelberg (1998)
Bernstein, D.J.: Arbitrarily tight bounds on the distribution of smooth integers. In: Bennett, Berndt, Boston, Diamond, Hildebrand, Philipp (eds.) Proceedings of the Millennial Conference on Number Theory, vol. 1, pp. 49–66. A. K. Peters (2002)
Bernstein, D.J.: Proving primality in essentially quartic time. Mathematics of Computation (to appear, 2006), http://cr.yp.to/papers.html#quartic
Brent, R.P.: Multiple precision zero-finding methods and the complexity of elementary function evaluation. In: Traub, J.F. (ed.) Analytic Computational Complexity, pp. 151–176. Academic Press, London (1976)
Canfield, E.R., Erdős, P., Pomerance, C.: On a problem of Oppenheim concerning Factorisatio Numerorum. Journal of Number Theory 17, 1–28 (1983)
Crandall, R., Pomerance, C.: Prime Numbers, a Computational Perspective. Springer, Heidelberg (2001)
de Bruijn, N.G.: On the number of positive integers ≤ x and free of prime factors > y. Indag. Math. 13, 50–60 (1951)
de Bruijn, N.G.: On the number of positive integers ≤ x and free of prime factors > y, II. Indag. Math. 28, 239–247 (1966)
Deléglise, M., Rivat, J.: Computing π(x): the Meissel, Lehmer, Lagarias, Miller, Odlyzko method. Math. Comp. 65(213), 235–245 (1996)
Hildebrand, A.: On the number of positive integers ≤ x and free of prime factors > y. Journal of Number Theory 22, 289–307 (1986)
Hildebrand, A., Tenenbaum, G.: On integers free of large prime factors. Trans. AMS 296(1), 265–290 (1986)
Hildebrand, A., Tenenbaum, G.: Integers without large prime factors. Journal de Théorie des Nombres de Bordeaux 5, 411–484 (1993)
Hildebrand, A.: On the local behavior of Ψ(x,y). Trans. Amer. Math. Soc. 297(2), 729–751 (1986)
Hunter, S., Sorenson, J.P.: Approximating the number of integers free of large prime factors. Mathematics of Computation 66(220), 1729–1741 (1997)
Knuth, D.E., Trabb Pardo, L.: Analysis of a simple factorization algorithm. Theoretical Computer Science 3, 321–348 (1976)
Menezes, A.J., van Oorschot, P.C., Vanstone, S.A.: Handbook of Applied Cryptography. CRC Press, Boca Raton (1997)
Moree, P.: Psixyology and Diophantine Equations. PhD thesis, Rijksuniversiteit Leiden (1993)
Norton, K.K.: Numbers with Small Prime Factors, and the Least kth Power Non-Residue. Memoirs of the American Mathematical Society, vol. 106. American Mathematical Society, Providence, Rhode Island (1971)
Pomerance, C. (ed.): Cryptology and Computational Number Theory. Proceedings of Symposia in Applied Mathematics, vol. 42. American Mathematical Society, Providence, Rhode Island (1990)
Schoenfeld, L.: Sharper bounds for the Chebyshev functions θ(x) and ψ(x). II. Mathematics of Computation 30(134), 337–360 (1976)
Sorenson, J.P.: A fast algorithm for approximately counting smooth numbers. In: Bosma, W. (ed.) ANTS 2000. LNCS, vol. 1838, pp. 539–549. Springer, Heidelberg (2000)
Suzuki, K.: An estimate for the number of integers without large prime factors. Mathematics of Computation 73, 1013–1022 (2004); MR 2031422 (2005a:11142)
Suzuki, K.: Approximating the number of integers without large prime factors. Mathematics of Computation 75, 1015–1024 (2006)
van de Lune, J., Wattel, E.: On the numerical solution of a differential-difference equation arising in analytic number theory. Mathematics of Computation 23, 417–421 (1969)
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Parsell, S.T., Sorenson, J.P. (2006). Fast Bounds on the Distribution of Smooth Numbers. In: Hess, F., Pauli, S., Pohst, M. (eds) Algorithmic Number Theory. ANTS 2006. Lecture Notes in Computer Science, vol 4076. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11792086_13
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DOI: https://doi.org/10.1007/11792086_13
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