Sur la concentration de certaines fonctions additives

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

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

Improving on estimates of Erdős, Halász and Ruzsa, we provide new upper and lower bounds for the concentration function of the limit law of certain additive arithmetic functions under hypotheses involving only their average behaviour on the primes. In particular we partially confirm a conjecture of Erdős and Kátai. The upper bound is derived via a reappraisal of the method of Diamond and Rhoads, resting upon the theory of functions with bounded mean oscillation.

Mots clés: additive functions; arithmetic functions; Berry– Esseen inequality; dis-tribution of arithmetic functions; concentration functions; probabilistic number theory; Erd˝ os-Wintner theorem; BMO functions

Domaines: Théorie des nombres [math.NT]

Régis de La Bretèche, Gerald Tenenbaum. Sur la concentration de certaines fonctions additives. Mathematical Proceedings of the Cambridge Philosophical Society, 2012, 152 (1), pp.179-189, 191. ⟨10.1017/S0305004111000521⟩. ⟨hal-01281334v2⟩