Oscillations localisées sur les diviseurs

Régis de La Bretèche (1) , Gerald Tenenbaum (2)

1 IMJ - Institut de Mathématiques de Jussieu
2 IECN - Institut Élie Cartan de Nancy

Let f be a real arithmetic function and ∆(n, f) denote the corresponding generalization of Hooley's Delta-function. We investigate weighted moments of ∆(n; f) for oscillating functions f, typical cases being those of a non principal Dirichlet character or of the Möbius function. We obtain, in particular, sharp bounds up to factors (log x) o(1) for all weighted finite integral, even moments computed on the integers not exceeding x. This is the key step to the proof, given in a subsequent work, of Manin's conjecture, in the strong form conjectured by Peyre and with an e↵ective remainder term, for all Châtelet surfaces. The proof of the main results rest upon a genuinely new approach for Hooley-type functions.

Mots clés: Hooley's Delta-function; divisors; Dirichlet characters; Möbius function; moments of arithmetic functions; concentration functions; Châtelet surfaces; Manin's conjecture; Peyre's

Domaines: Théorie des nombres [math.NT]

Régis de La Bretèche, Gerald Tenenbaum. Oscillations localisées sur les diviseurs. Journal of the London Mathematical Society, 2012, 85, pp.669-693. ⟨hal-01281336⟩