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 $$H_{\gamma }(x)\sim C(\gamma )x^{1/2+\gamma }/\log x,x\to \infty ,$$ where $C(\gamma )>0$ is an explicit constant depending only on $\gamma$. This extends the previous result of {É}.~Fouvry (2013), which has $12/25$ instead of $9/20$.

12 pages

References in corpus (1)

• Algebraic trace functions over the primes