Dirichlet series and hyperelliptic curves.

For a fixed hyperelliptic curve C given by the equation y² = f( x) with f ∈ ℤ[ x] having distinct roots and degree at least 5, we study the variation of rational points on the quadratic twists C_m whose equation is given by my² = f( x). More precisely, we study the Dirichlet series where the summation is over all non-zero squarefree integers. We show that converges for ℜ( s) > 1. We extend its range of convergence assuming the ABC conjecture. This leads us to study related Dirichlet series attached to binary forms. We are then led to investigate the variation of rational points on twists of superelliptic curves. We apply this study to certain classical problems of analytic number theory such as the number of powerfree values of a fixed polynomial in ℤ[ x]. [ABSTRACT FROM AUTHOR]