PreprintCC BY 4.0arXiv
On a Turán's theorem for small primes
Abstract
Denote by ω(n) the number of distinct prime divisors of the natural number n. In 2007, Granville and Soundararajan gave a quite new method to compute the higher moments ∑n≤ x(ω(n)−loglog x)k, for a wide range of integers k≥ 2. In this notes, we shall apply the method for ωz(n) which denotes the number of distinct prime divisors ≤ z of n. Especially, for odd integers k≥ 3, we lead asymptotic formulas for ∑n≤ x(ωz(n)−loglog z)k, under certain restrictions on k, z, and x. Also, we refer to a framework of the approach.
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BibTeX
@misc{minamide2026turáns,
title = {{On a Turán's theorem for small primes}},
author = {T. Makoto Minamide and Haruka Sakai and Yoshio Tanigawa},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.22777},
eprint = {2609.22777},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.22777},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - On a Turán's theorem for small primes
AU - T. Makoto Minamide
AU - Haruka Sakai
AU - Yoshio Tanigawa
PY - 2026
DA - 2026/09/19
PB - arXiv
DO - 10.48550/arxiv.2609.22777
UR - https://arxiv.org/abs/2609.22777
AB - Denote by ω(n) the number of distinct prime divisors of the natural number n. In 2007, Granville and Soundararajan gave a quite new method to compute the higher moments ∑_n≤ x(ω(n)−loglog x)^k, for a wide range of integers k≥ 2. In this notes, we shall apply the method for ω_z(n) which denotes the number of distinct prime divisors ≤ z of n. Especially, for odd integers k≥ 3, we lead asymptotic formulas for ∑_n≤ x(ω_z(n)−loglog z)^k, under certain restrictions on k, z, and x. Also, we refer to a framework of the approach.
ER -