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On the square-free sieve. (English) Zbl 1057.11043

The main result of this paper is an improvement in the error terms in the known estimates for the occurrence of squarefree numbers among the values of integer polynomials in a single variable, or among the values of homogeneous integer polynomials in two variables.
When \(f(x)\) is an irreducible integer polynomial of degree 3, one seeks to show that the number of positive integers \(x \leq N\) for which \(F(x)\) is squarefree is estimated as \(C_ f N+O\bigl(N/\!\log^\beta\!N\bigr)\), for a certain constant \(C_ f\), positive provided \(F(x)\) has no fixed square divisor. C. Hooley [Applications of sieve methods to the theory of numbers. Cambridge Tracts in Mathematics 70, Cambridge, Univ. Press (1976: Zbl 0327.10044)] obtained such a result with \(\beta={1\over2}\), using the large sieve. By also invoking considerations relating to elliptic curves the author improves this exponent to \(\beta= 0.6829\ldots\,\), and to \(\beta=0.8061\ldots\) if the discriminant of \(f\) is a square. These exponents arise as values of logarithms of specified rational functions of \(\sin {1\over3}\pi\).
When \(f(x,y)\) is homogeneous, irreducible and of degree 6 the reviewer [Q. J. Math., Oxf.II. Ser.43, No. 169, 45–65 (1992: Zbl 0768.11034)] showed that the number of pairs \(x,y\) with \(| x| \leq N\), \(| y| \leq N\) and \(f(x,y)\) squarefree can be similarly estimated as \(C_ f N^ 2 +O( N^ 2/\!\log^ \beta\!N)\). The reviewer’s value of \(\beta\) was \({1\over3}\), but K. Ramsay pointed out, in a private communication made shortly after the submission of the paper, that this value could be improved to \(1\over2\) by using an observation also described in the paper currently under review. The author’s values of \(\beta\) vary between \(0.8309\ldots\) and \(0.7188\), depending on the Galois group of the splitting field of the equation \(f(x,1)=0\).
When \(f(x,y)\) has degree \(<6\) the author improves the \(O\)-term to \(O\bigl(N^ {4/3}\log N\bigr)\) if the degree \(d\) satisfies \(d \leq 4\), and to \(O\bigl(N^ {c+\varepsilon}\bigr)\) with \(c=(5+\sqrt{113})/8\) if \(d=5\). As in the earlier work, the relevant parameter \(d\) is actually the largest degree of an irreducible factor of the (possibly reducible) polynomial \(f\).
Other consequences of the author’s work involve the distribution of root numbers of elliptic curves. Let \(f(x)\) be an irreducible polynomial of degree 6. For squarefree \(d\) let \(C_ d\) be the hyperelliptic curve \(dy^ 2=f(x)\), and denote \(R(\alpha,d) =2^{\alpha\,\text{ rank}(C_ d)}\). Then the author obtains estimates of the shape \(\sum_ {d<X} R(\alpha,d) \ll X\log^ {\delta-1}\!X\), where \(\delta\) depends in an explicit way on \(\alpha\) and the Galois group of the splitting field of \(f\).
The authors develops some of the tools employed in greater generality than is actually needed in the current paper, but which may possibly find application on a subsequent occasion.

MSC:

11N32 Primes represented by polynomials; other multiplicative structures of polynomial values
11N36 Applications of sieve methods
11G05 Elliptic curves over global fields
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