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Power-free values of binary forms. (English) Zbl 0768.11034

C. Hooley [Mathematika 14, 21–26 (1967; Zbl 0166.05203)]established an asymptotic formula for the number of \((d-1)\)-free integers \(n\leq x\) represented by an irreducible polynomial of degree \(d\geq 3\) having no fixed \((d-1)\)th power divisors. F. Gouvêa and B. Mazur [J. Am. Math. Soc. 4, 1–23 (1991; Zbl 0725.11027)] adapted Hooley’s method to the problem of showing that a positive proportion of the values assumed by a binary cubic form \(f(a,b)\) are squarefree. They proceeded by fixing \(a\) and showing that the resulting estimate held with sufficient uniformity to draw the desired conclusion.
In this paper, the author treats \(f(a,b)\) in a binary way and thereby obtains much stronger conclusions. In particular, he obtains an asymptotic formula for the number of \(a,b\leq x\) such that \(f(a,b)\) is squarefree when \(f\) is a suitable binary form of degree not exceeding \(6\). When \(k\geq 3\) and \(f\) is a suitable binary form of degree \(\leq 2k+1\), he obtains another asymptotic formula for the number of \(a,b\leq x\) such that \(f(a,b)\) is \(k\)-free.

MSC:

11N32 Primes represented by polynomials; other multiplicative structures of polynomial values
11E16 General binary quadratic forms
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