Primary components and large prime factors of cyclotomic values
Abstract
Let Φr(a,b) denote the homogeneous cyclotomic value. For sufficiently large A, uniformly for 1/2≤ B≤ A, r≥3 and Z≥2, the number of coprime pairs in (A,2A]×(B,2B] whose greatest prime factor P(Φr(a,b)) is at most Z is O(AB/φ(r)+√(r/φ(r)) π(Z;r,1)), where π(Z;r,1) counts primes at most Z congruent to $1$ modulo r. If n=n(A)→∞ through positive integers and 1≤ H(A)=o(log A), then P(a2n+1)>2nAH(A) for almost all a∈(A,2A], and P(a2n+b2n)>2nA2H(A) for almost all coprime pairs in (A,2A]2, with no further restriction on n. For every fixed α>0 and every r≥ Aα, pairwise distinct primitive prime divisors of Φr(a) can be selected for all but Oα(A1−cα) bases in (A,2A], where cα>0. For coprime positive integers a,b with max(a,b)≥2 and r≥3, put C=Φr(a,b) and S=φ(r)2logmax(a,b). Uniformly as S→∞, maxp| Cpvp(C)≥(5/16−o(1))S. For every positive function h(S)→∞, ∑p| C, pvp(C)>S/h(S)vp(C)log p =(1−o(1))log C uniformly in the same limit.
Related
- Cyclotomic polynomials with trinomial irreducible factors over finite fields
- Invariant factors from products of cyclotomic polynomials, with applications
- Cyclotomic factors and irreducibility of the denominators of q-deformed rational numbers
- Large values of cubic and quartic Dirichlet L-functions of prime moduli
- Large values of quadratic Hecke L-functions with prime-related moduli of the Gaussian field
BibTeX
@misc{xie2026primary,
title = {{Primary components and large prime factors of cyclotomic values}},
author = {Wei Xie},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.00075},
eprint = {2611.00075},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.00075},
}
RIS
TY - UNPB
TI - Primary components and large prime factors of cyclotomic values
AU - Wei Xie
PY - 2026
DA - 2026/11/01
PB - arXiv
DO - 10.48550/arxiv.2611.00075
UR - https://arxiv.org/abs/2611.00075
AB - Let Φ_r(a,b) denote the homogeneous cyclotomic value. For sufficiently large A, uniformly for 1/2≤ B≤ A, r≥3 and Z≥2, the number of coprime pairs in (A,2A]×(B,2B] whose greatest prime factor P(Φ_r(a,b)) is at most Z is O(AB/φ(r)+√(r/φ(r)) π(Z;r,1)), where π(Z;r,1) counts primes at most Z congruent to $1$ modulo r. If n=n(A)→∞ through positive integers and 1≤ H(A)=o(log A), then P(a^2^n+1)>2^nAH(A) for almost all a∈(A,2A], and P(a^2^n+b^2^n)>2^nA^2H(A) for almost all coprime pairs in (A,2A]^2, with no further restriction on n. For every fixed α>0 and every r≥ A^α, pairwise distinct primitive prime divisors of Φ_r(a) can be selected for all but O_α(A^1−c_α) bases in (A,2A], where c_α>0. For coprime positive integers a,b with max(a,b)≥2 and r≥3, put C=Φ_r(a,b) and S=φ(r)^2logmax(a,b). Uniformly as S→∞, max_p| Cp^v_p(C)≥(5/16−o(1))S. For every positive function h(S)→∞, ∑_p| C, p^v_p(C)>S/h(S)v_p(C)log p =(1−o(1))log C uniformly in the same limit.
ER -