On a generalization of Beiter Conjecture

Abstract

We prove that for every $\varepsilon>0$ and a nonnegative integer $ω$ there exist primes $p_1,p_2,\ldots,p_ω$ such that for $n=p_1p_2\ldots p_ω$ the height of the cyclotomic polynomial $Φ_n$ is at least $(1-\varepsilon)c_ωM_n$, where $M_n=\prod_{i=1}^{ω-2}p_i^{2^{ω-1-i}-1}$ and $c_ω$ is a constant depending only on $ω$; furthermore $\lim_{ω\to\infty}c_ω^{2^{-ω}}\approx0.71$. In our construction we can have $p_i>h(p_1p_2\ldots p_{i-1})$ for all $i=1,2,\ldots,ω$ and any function $h:\mathbb{R}_+\to\mathbb{R}_+$.