Exponential sums and additive problems involving square-free numbers

• Jörg Brüdern ^[1] ; Alberto Perelli ^[2]
  1. [1] University of Stuttgart

     University of Stuttgart

     Stadtkreis Stuttgart, Alemania

     
  2. [2] Universitá di Genova, Italia
• Localización: Annali della Scuola Normale Superiore di Pisa. Classe di scienze, ISSN 0391-173X, Vol. 28, Nº 4, 1999, págs. 591-613
• Idioma: inglés
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• Resumen

  • Let r v (N) denote the number of representations of the integer N as a sum of v square-free numbers. We obtain unconditional and conditional bounds for the error term in the asymptotic formula for rv (N), when v > 3. The conditional bounds are essentially best possible for v > 4. The unconditional bounds are, for v > 3, essentially best possible with respect to the present knowledge on the distribution of the zeros of the Riemann zeta function. Proofs are based on the circle method. The main ingredients are a new pointwise estimate for the exponential sum S(a) over square-free numbers and a recent bound (see [3]) for the L2-norm of S(a) restricted to the minor arcs.