On $rth$ coefficient of divisors of $x^n-1$

Abstract

Let $r,n$ be two natural numbers and let $H(r,n)$ denote the maximal absolute value of $r$th coefficient of divisors of $x^n-1$. In this paper, we show that $\sum_{n\leq x}H(r,n)$ is asymptotically equal to $c(r)x(\log x)^{2^r-1}$ for some constant $c(r)>0$. Furthermore, we give an explicit expression of $c(r)$ in terms of $r$.