JournalscmhVol. 87, No. 3pp. 639–667

Friable values of binary forms

  • Antal Balog

    Hungarian Academy, Budapest, Hungary
  • Valentin Blomer

    Georg-August-Universität Göttingen, Germany
  • Cécile Dartyge

    Université Henri Poincaré, Vandoeuvre lès Nancy, France
  • Gérald Tenenbaum

    Université de Lorraine, Vandoeuvre-lès-Nancy, France
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Abstract

Let FZ[X,Y]F \in \mathbb{Z}[X, Y] be an integral binary form of degree g2g\geq 2, and let

ΨF(x,y):=card{1a,bx:P+(F(a,b))y}\Psi_F(x, y) := \mathrm{card}\{1 \leqslant a, b \leqslant x : P^{+}(F(a, b)) \leqslant y \}

where as usual P+(n)P^{+}(n) denotes the largest prime factor of nn. It is proved that ΨF(x,y)x2\Psi_F(x, y) \asymp x^2 for y=xg2+εy = x^{g-2 +\varepsilon} in general, and y=x1/e+εy=x^{1/\sqrt{\rm e}+\varepsilon} if g=3g=3. Better results are obtained if FF is reducible.

Cite this article

Antal Balog, Valentin Blomer, Cécile Dartyge, Gérald Tenenbaum, Friable values of binary forms. Comment. Math. Helv. 87 (2012), no. 3, pp. 639–667

DOI 10.4171/CMH/264