On Nonzero Coefficients of Binary Cyclotomic Polynomials

2023/07/14 by Igor E. Shparlinski, Shparlinski, Igor E., Laurence P. Wijaya +1
Mathematics · #11B83 #11L07 #Analytic Number Theory Research #FOS: Mathematics #Mathematical functions and polynomials #Meromorphic and Entire Functions #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2307.07229

openalex publication_date 2023/07/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let ϑ(m) is number of nonzero coefficients in the m-th cyclotomic polynomial. For real γ> 0 and x ≥ 2 we define H_γ(x)=#\m:~m=pq ≤ x, p 0, uniformly over γ with9/20+η≤ γ≤ 1/2 -η, we have an asymptotic formula H_γ(x)∼ C(γ)x^1/2+γ/ log x, x → ∞, where C(γ)> 0 is an explicit constant depending only on γ. This extends the previous result of É.~Fouvry (2013), which has 12/25 instead of 9/20.

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