On the friable mean-value of the Erdős–Hooley Delta function

B. Martin (1) , G. Tenenbaum (2) , J. Wetzer (1)

1 LMPA - Laboratoire de Mathématiques Pures et Appliquées Joseph Liouville
2 IECL - Institut Élie Cartan de Lorraine

For integer n and real u, define ∆(n, u) := |{d : d | n, e^u < d ⩽ e^(u+1) }|. Then, the Erdős-Hooley Delta function is defined as ∆(n) := max_{u∈R} ∆(n, u). We provide uniform upper and lower bounds for the mean-value of ∆(n) over friable integers, i.e. integers free of large prime factors.

Mots clés: friable integers; mean-value of arithmetic functions; Erdős-Hooley Delta function; saddle-point method

Domaines: Mathématiques [math]

B. Martin, G. Tenenbaum, J. Wetzer. On the friable mean-value of the Erdős–Hooley Delta function. Indagationes Mathematicae, 2024, 35 (2), pp.376-389. ⟨10.1016/j.indag.2024.02.002⟩. ⟨hal-04544245⟩