On Nonzero Coefficients of Binary Cyclotomic Polynomials

TL;DR

This paper investigates the distribution of nonzero coefficients in binary cyclotomic polynomials, deriving an asymptotic formula for the count of such polynomials with certain properties, extending previous results by Fouvry.

Contribution

It provides a new asymptotic estimate for the number of binary cyclotomic polynomials with bounded nonzero coefficients, improving the range of parameters compared to prior work.

Findings

Asymptotic formula for the count of binary cyclotomic polynomials with bounded nonzero coefficients.

Extension of Fouvry's 2013 result to a wider parameter range.

Explicit constant depending on gamma in the asymptotic expression.

Abstract

Let $\vartheta(m)$ is number of nonzero coefficients in the $m$-th cyclotomic polynomial. For real $\gamma > 0$ and $x \ge 2$ we define $$H_{\gamma}(x)=\#\left\{m:~m=pq \le x, \ p<q\text{ primes }, \ \vartheta(m)\le m^{1/2+\gamma}\right\}, $$ and show that for any fixed $\eta> 0$, uniformly over $\gamma$ with $$9/20+\eta \le \gamma\le 1/2 -\eta, $$ we have an asymptotic formula $$ H_{\gamma}(x)\sim C(\gamma)x^{1/2+\gamma}/ \log x, \qquad x \to \infty, $$ where $C(\gamma)> 0$ is an explicit constant depending only on $\gamma$. This extends the previous result of {\'E}.~Fouvry (2013), which has $12/25$ instead of $9/20$.