Cyclotomic polynomials with prescribed height and prime number theory

Abstract

Given any positive integer $n,$ let $A(n)$ denote the height of the $n^{\text{th}}$ cyclotomic polynomial, that is its maximum coefficient in absolute value. It is well known that $A(n)$ is unbounded. We conjecture that every natural number can arise as value of $A(n)$ and prove this assuming that for every pair of consecutive primes $p$ and $p'$ with $p\ge 127$ we have $p'-p<\sqrt{p}+1.$ We also conjecture that every natural number occurs as maximum coefficient of some cyclotomic polynomial and show that this is true if Andrica's conjecture that always $\sqrt{p'}-\sqrt{p}<1$ holds. This is the first time, as far as the authors know, a connection between prime gaps and cyclotomic polynomials is uncovered. Using a result of Heath-Brown on prime gaps we show unconditionally that every natural number $m\le x$ occurs as $A(n)$ value with at most $O_{\epsilon}(x^{3/5+\epsilon})$ exceptions. On the Lindel\"of Hypothesis we show there are at most $O_{\epsilon}(x^{1/2+\epsilon})$ exceptions and study them further by using deep work of Bombieri--Friedlander--Iwaniec on the distribution of primes in arithmetic progressions beyond the square-root barrier.