On nonzero coefficients of binary cyclotomic polynomials

Abstract

Let $\vartheta \left(m\right)$ be the number of nonzero coefficients in the m-th cyclotomic polynomial. For real $\gamma >0$ and $x\ge 2$ we define$H_{\gamma }\left(x\right)=\#\{m:m=pq\le x,p<q\text{ primes },\vartheta (m)\le m^{1/2+\gamma }\},$ and show that for any fixed $\eta >0$, uniformly over γ with$9/20+\eta \le \gamma \le 1/2-\eta ,$ we have an asymptotic formula$H_{\gamma }\left(x\right)\sim C\left(\gamma \right)x^{1/2+\gamma }/\log x,x\to \infty ,$ where $C\left(\gamma \right)>0$ 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. This improvement is based on new ingredient including work of W. Duke, J. Friedlander and H. Iwaniec (1997).