Report
An Algorithm for Ennola's Second Theorem and Counting Smooth Numbers in Practice
| العنوان: | An Algorithm for Ennola's Second Theorem and Counting Smooth Numbers in Practice |
|---|---|
| المؤلفون: | Makdad, Chloe, Sorenson, Jonathan P. |
| سنة النشر: | 2022 |
| المجموعة: | Computer Science Mathematics |
| مصطلحات موضوعية: | Mathematics - Number Theory, Computer Science - Data Structures and Algorithms, 11Y16, 11Y05 |
| الوصف: | Let $\Psi(x,y)$ count the number of positive integers $n\le x$ such that every prime divisor of $n$ is at most $y$. Given inputs $x$ and $y$, what is the best way to estimate $\Psi(x,y)$? We address this problem in three ways: with a new algorithm to estimate $\Psi(x,y)$, with a performance improvement to an established algorithm, and with empirically based advice on how to choose an algorithm to estimate $\Psi$ for the given inputs. Our new algorithm to estimate $\Psi(x,y)$ is based on Ennola's second theorem [Ennola69], which applies when $y< (\log x)^{3/4-\epsilon}$ for $\epsilon>0$. It takes $O(y^2/\log y)$ arithmetic operations of precomputation and $O(y\log y)$ operations per evaluation of $\Psi$. We show how to speed up Algorithm HT, which is based on the saddle-point method of Hildebrand and Tenenbaum [1986], by a factor proportional to $\log\log x$, by applying Newton's method in a new way. And finally we give our empirical advice based on five algorithms to compute estimates for $\Psi(x,y)$.The challenge here is that the boundaries of the ranges of applicability, as given in theorems, often include unknown constants or small values of $\epsilon>0$, for example, that cannot be programmed directly. |
| نوع الوثيقة: | Working Paper |
| URL الوصول: | http://arxiv.org/abs/2208.01725 |
| رقم الانضمام: | edsarx.2208.01725 |
| قاعدة البيانات: | arXiv |
| الوصف غير متاح. |