TY - JOUR
T1 - Squarefree density of polynomials II
AU - Kowalski, J. M.
AU - Vaughan, R. C.
N1 - Publisher Copyright:
© The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature 2024.
PY - 2024
Y1 - 2024
N2 - In this memoir we study the density of squarefree numbers in the set of numbers of the form (Formula presented.) where b, c are a non-zero integers with (b,c)=1 and k≥2. Suppose that θ is a constant with 0<θ≤3/k, that X is large, and Y≍Xθ. Then take NP(X,Y) to be the number of pairs of integers x and y with |x|≤X, |y|≤Y such that P(x,y) is squarefree. Suppose also that ρP(d) is the number of solutions of P(x,y)≡0(modd). Then we show that (Formula presented.) where SP is the anticipated density (Formula presented.) When k≥5 this is new.
AB - In this memoir we study the density of squarefree numbers in the set of numbers of the form (Formula presented.) where b, c are a non-zero integers with (b,c)=1 and k≥2. Suppose that θ is a constant with 0<θ≤3/k, that X is large, and Y≍Xθ. Then take NP(X,Y) to be the number of pairs of integers x and y with |x|≤X, |y|≤Y such that P(x,y) is squarefree. Suppose also that ρP(d) is the number of solutions of P(x,y)≡0(modd). Then we show that (Formula presented.) where SP is the anticipated density (Formula presented.) When k≥5 this is new.
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U2 - 10.1007/s00208-024-03069-3
DO - 10.1007/s00208-024-03069-3
M3 - Article
AN - SCOPUS:85213501460
SN - 0025-5831
JO - Mathematische Annalen
JF - Mathematische Annalen
ER -