G\'erald TenenbaumAffiliation: Institut Elie Cartan Nancy Email: gerald.tenenbaum@univ-lorraine.fr Title Of Talk: On the core of an integer Abstract: The core, or squarefree kernel, of an integer is defined as its largest squarefree divisor. The underlying arithmetic function is linked to many problems and conjectures in Number Theory, including the $abc$ conjecture. We shall present new, uniform asymptotic formulae for the number of integers not exceeding $x$ with core at most $y$. These estimates have been obtained recently in collaboration with O. Robert. The talk will also describe a number of applications to questions studied by Erd{\H o}s, de Bruijn and others. We will finally provide an account on how these bounds are employed, in a joint paper with O. Robert and C.L. Stewart, to get a refined form of the $abc$ conjecture. WARNING: This page contains MATH-JAX
Last update made Tue Oct 23 09:20:28 EDT 2012.
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