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

TL;DR

This paper investigates the maximal absolute value of the r-th coefficient of divisors of x^n-1, establishing an asymptotic formula for their sum and providing an explicit expression for the constant involved.

Contribution

It introduces an asymptotic formula for the sum of maximal coefficients of divisors of x^n-1 and derives an explicit expression for the constant c(r).

Findings

Sum of H(r,n) over n ≤ x is asymptotically c(r) x (log x)^{2^r - 1}

Explicit formula for c(r) in terms of r

Provides new insights into the behavior of coefficients of divisors of cyclotomic polynomials

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 $r$.