On Nonzero Coefficients of Binary Cyclotomic Polynomials

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 $9/20$.