An L2-BOUND FOR THE BARBAN–VEHOV WEIGHTS

: Ramaré, Olivier; Zuniga-Alterman, Sebastian

Publisher: Adam Mickiewicz University (Euclid)

: 2026

Functiones Et Approximatio: Commentarii Mathematici

: 74

: 1

: 74

: 80

: 0208-6573

: 2080-9433

DOI: https://doi.org/10.7169/facm/241018-19-5

: https://doi.org/10.7169/facm/241018-19-5

Let λ be the Barban–Vehov weights, defined in (1). Let X ≥ z₁ ≥ 100 and z₂ = z₁². We prove that

∑{n ≤ X} (1/n) (∑{d | n} λ_d)² ≤ 30 · (log X) / log(z₂ / z₁),

saving more than a factor of 5 on what was the best known constant in such an inequality. Two related estimates are also provided for X ≥ z₁ ≥ 100 and z₂ = z₁^τ for some τ > 1.

:
The second author has been supported by the Finnish Centre of Excellence in Randomness and Structures grant 346307 of the Academy of Finland