PreprintCC BY 4.0arXiv
On a Turán's theorem for arithmetic progressions
Abstract
Let m≥ 1 be a fixed integer, a an integer satisfying (a,m)=1, and z≥ 1 a real parameter. Denote by ωz(n;m,a) the number of distinct prime divisors p of n satisfying p≡ a (m) and p≤ z. We study an asymptotic behaviour of ∑n≤ x(ωz(n;m,a)−1/φ(m)loglog z)k as x→∞ for a wide range of positive integer k≥ 2, where φ(·) is the Euler function. Following a method of Granville and Soundararajan we lead an asymptotic formula for the above. Also, we investigate ∑n≤ x(ω(n;m,a)−1/φ(m)loglog x)k, where ω(n;m,a) denotes the number of distinct prime divisors p of n such that p≡ a (m).
Related
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BibTeX
@misc{minamide2026turáns,
title = {{On a Turán's theorem for arithmetic progressions}},
author = {T. Makoto Minamide and Haruka Sakai and Yoshio Tanigawa},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.22784},
eprint = {2609.22784},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.22784},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - On a Turán's theorem for arithmetic progressions
AU - T. Makoto Minamide
AU - Haruka Sakai
AU - Yoshio Tanigawa
PY - 2026
DA - 2026/09/19
PB - arXiv
DO - 10.48550/arxiv.2609.22784
UR - https://arxiv.org/abs/2609.22784
AB - Let m≥ 1 be a fixed integer, a an integer satisfying (a,m)=1, and z≥ 1 a real parameter. Denote by ω_z(n;m,a) the number of distinct prime divisors p of n satisfying p≡ a (m) and p≤ z. We study an asymptotic behaviour of ∑_n≤ x(ω_z(n;m,a)−1/φ(m)loglog z)^k as x→∞ for a wide range of positive integer k≥ 2, where φ(·) is the Euler function. Following a method of Granville and Soundararajan we lead an asymptotic formula for the above. Also, we investigate ∑_n≤ x(ω(n;m,a)−1/φ(m)loglog x)^k, where ω(n;m,a) denotes the number of distinct prime divisors p of n such that p≡ a (m).
ER -