2004年4月
An estimate for the number of integers without large prime factors
Mathematics of Computation
- 巻
- 73
- 号
- 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 >
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 >
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 <
log y <
(log x)/e, where ε is any positive constant.
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 >
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 <
log y <
(log x)/e, where ε is any positive constant.
- ID情報
-
- DOI : 10.1090/S0025-5718-03-01571-0
- ISSN : 0025-5718
- SCOPUS ID : 1642602844