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THE DISTRIBUTION OF DIVISORS OF POLYNOMIALS
Author(s) -
Ford Kevin,
Qian Guoyou
Publication year - 2020
Publication title -
mathematika
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.955
H-Index - 29
eISSN - 2041-7942
pISSN - 0025-5793
DOI - 10.1112/mtk.12033
Subject(s) - mathematics , divisor (algebraic geometry) , integer (computer science) , combinatorics , order (exchange) , degree (music) , greatest common divisor , distribution (mathematics) , irreducible polynomial , divisor function , polynomial , discrete mathematics , mathematical analysis , physics , finance , matrix polynomial , computer science , acoustics , economics , programming language
Let F ( x ) be an irreducible polynomial with integer coefficients and degree at least 2. For x ⩾ z ⩾ y ⩾ 2 , denote byH F ( x , y , z )the number of integers n ⩽ x such that F ( n ) has at least one divisor d with y < d ⩽ z . We determine the order of magnitude ofH F ( x , y , z )uniformly for y + y / log C y < z ⩽ y 2and y ⩽ x 1 − δ, showing that the order is the same as the order of H ( x , y , z ) , the number of positive integers n ⩽ x with a divisor in ( y , z ] . Here C is an arbitrarily large constant and δ > 0 is arbitrarily small.