On Nonzero Coefficients of Binary Cyclotomic Polynomials

Abstract

Let $\vartheta(m)$ is number of nonzero coefficients in the $m$-th cyclotomic polynomial. For real $γ> 0$ and $x \ge 2$ we define $$H_γ(x)=\#\left\{m:~m=pq \le x, \ p 0$, uniformly over $γ$ with $$9/20+η\le γ\le 1/2 -η, $$ we have an asymptotic formula $$ H_γ(x)\sim C(γ)x^{1/2+γ}/ \log x, \qquad x \to \infty, $$ 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 $9/20$.