論文

査読有り 筆頭著者 責任著者
2004年4月

An estimate for the number of integers without large prime factors

Mathematics of Computation
  • Koji Suzuki

73
246
開始ページ
1013
終了ページ
1022
記述言語
英語
掲載種別
研究論文(学術雑誌)
DOI
10.1090/S0025-5718-03-01571-0

Ψ(x, y) denotes the number of positive integers ≤ x and free of prime factors &gt
y. Hildebrand and Tenenbaum provided a good approximation of ψ(x,y). However, their method requires the solution α = α(x,y) to the equation Σp≤ylog p/(pα-1) = log x, and therefore it needs a large amount of time for the numerical solution of the above equation for large y. Hildebrand also showed 1 - ξu/log y approximates α for 1 ≤ u ≤ y/(2 log y), where u=(log x)/log y and ξu is the unique solution to eξu = 1 + uξ u. Let E(i) be defined by E(0) = log u
E(i) = log u + log(E(i - 1) + 1/u) for i &gt
0. We show E(m) approximates ξu, and 1 - E(m)/log y also approximates α, where m = [(log u+log log y)/ log log u] + 1. Using these approximations, we give a simple method which approximates Ψ(x, y) within a factor 1 + O(1/u + 1/log y) in the range (log log x) 5/3+e &lt
log y &lt
(log x)/e, where ε is any positive constant.

リンク情報
DOI
https://doi.org/10.1090/S0025-5718-03-01571-0
ID情報
  • DOI : 10.1090/S0025-5718-03-01571-0
  • ISSN : 0025-5718
  • SCOPUS ID : 1642602844

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