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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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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