Large absolute values of cyclotomic polynomials at roots of unity

Abstract

The nth cyclotomic polynomial Φn(X) is the minimal polynomial of ζn:=e2πi/n. Given an integer m≥1 and a prescribed set S of arithmetic progressions modulo m, we define nx as the product of the primes p≤x lying in those progressions. Let d(n) denote the number of divisors of n. It turns out that under certain conditions on S and m there exists jx such that log⁡|Φnx(ζmjx)|/d(nx) tends to a positive limit. Our aim is to determine those conditions. We use the arithmetic of cyclotomic number fields, non-standard properties of character tables of finite abelian groups and a recent theorem of Bzdȩga, Herrera-Poyatos and Moree. After developing some generalities, we restrict to the case where m is a prime.Our motivation comes from a paper of Vaughan (1975). He studied the case where S={±2(mod5)} and used it to show that the maximum coefficient in absolute value of Φn can be very large.