On the unimodal character of the frequency function of the largest prime factor

Jean-Marie De Koninck, and Jason Pierre Sweeney. "On the unimodal character of the frequency function of the largest prime factor." Colloquium Mathematicae 88.2 (2001): 159-174. <http://eudml.org/doc/283852>.

@article{Jean2001,
abstract = {The main objective of this paper is to analyze the unimodal character of the frequency function of the largest prime factor. To do that, let P(n) stand for the largest prime factor of n. Then define f(x,p): = #\{n ≤ x | P(n) = p\}. If f(x,p) is considered as a function of p, for 2 ≤ p ≤ x, the primes in the interval [2,x] belong to three intervals I₁(x) = [2,v(x)], I₂(x) = ]v(x),w(x)[ and I₃(x) = [w(x),x], with v(x) < w(x), such that f(x,p) increases for p ∈ I₁(x), reaches its maximum value in I₂(x), in which interval it oscillates, and finally decreases for p ∈ I₃(x). In fact, we show that v(x) ≥ √(log x) and w(x) ≤ √x. We also provide several conditions on primes p ≤ q so that f(x,p) ≥ f(x,q).},
author = {Jean-Marie De Koninck, Jason Pierre Sweeney},
journal = {Colloquium Mathematicae},
language = {eng},
number = {2},
pages = {159-174},
title = {On the unimodal character of the frequency function of the largest prime factor},
url = {http://eudml.org/doc/283852},
volume = {88},
year = {2001},
}

TY - JOUR
AU - Jean-Marie De Koninck
AU - Jason Pierre Sweeney
TI - On the unimodal character of the frequency function of the largest prime factor
JO - Colloquium Mathematicae
PY - 2001
VL - 88
IS - 2
SP - 159
EP - 174
AB - The main objective of this paper is to analyze the unimodal character of the frequency function of the largest prime factor. To do that, let P(n) stand for the largest prime factor of n. Then define f(x,p): = #{n ≤ x | P(n) = p}. If f(x,p) is considered as a function of p, for 2 ≤ p ≤ x, the primes in the interval [2,x] belong to three intervals I₁(x) = [2,v(x)], I₂(x) = ]v(x),w(x)[ and I₃(x) = [w(x),x], with v(x) < w(x), such that f(x,p) increases for p ∈ I₁(x), reaches its maximum value in I₂(x), in which interval it oscillates, and finally decreases for p ∈ I₃(x). In fact, we show that v(x) ≥ √(log x) and w(x) ≤ √x. We also provide several conditions on primes p ≤ q so that f(x,p) ≥ f(x,q).
LA - eng
UR - http://eudml.org/doc/283852
ER -