Pré-Publication, Document De Travail Année : 2026

On the concentration of divisors of powers

Sur la concentration des diviseurs des puissances

Résumé

For integer n≥ 1 and real u, let ∆(n,u):=|{d:d | n, \e^u < d≤ \e^{u+1}}|. The Erdős--Hooley Delta-function is then defined by ∆(n):=\max_{u \in R}∆(n,u).$ For any fixed integer r ≥ 2 and any irreducible polynomial F\in Z[X], we estimate the normal order of ∆(|F(n)|^r) to within factors that are slowly varying functions of log n. This is then applied to determine with the same precision the normal order of ∆(|F(n)|) for any polynomial F with integer coefficients. We also evaluate some weighted average orders of ∆(n^r) and apply the result to bounding short sums of ∆(|F(n)|) for general F\in Z[X].

Fichier principal
Vignette du fichier
Delta(n^r).pdf (389.09 Ko) Télécharger le fichier

Dates et versions

hal-05530884 , version 1 (28-02-2026)
hal-05530884 , version 2 (02-03-2026)
hal-05530884 , version 3 (10-03-2026)
hal-05530884 , version 4 (13-03-2026)
hal-05530884 , version 5 (14-04-2026)
hal-05530884 , version 6 (21-04-2026)
hal-05530884 , version 7 (07-05-2026)
hal-05530884 , version 8 (12-05-2026)
hal-05530884 , version 9 (15-05-2026)

Licence

Identifiants

  • HAL Id : hal-05530884 , version 9

Citer

Régis de la Bretèche, Tenenbaum Gérald. On the concentration of divisors of powers. 2026. ⟨hal-05530884v9⟩
250 Consultations
170 Téléchargements

Partager

  • More