Computing &pgr;(x): the Meissel, Lehmer, Lagarias, Miller, Odlyzko method
Mathematics of Computation
Approximating the number of integers free of large prime factors
Mathematics of Computation
Handbook of Applied Cryptography
Handbook of Applied Cryptography
ANTS-III Proceedings of the Third International Symposium on Algorithmic Number Theory
A Fast Algorithm for Appoximately Counting Smooth Numbers
ANTS-IV Proceedings of the 4th International Symposium on Algorithmic Number Theory
Approximately counting semismooth integers
Proceedings of the 38th international symposium on International symposium on symbolic and algebraic computation
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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 y2/3. 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.