On a generalization of Beiter Conjecture

Abstract

We prove that for every $\epsilon >0$ and a nonnegative integer $\omega$ there exist primes $p_1,p_2,\dots ,p_{\omega }$ such that for $n=p_1{p}_2\dots p_{\omega }$ the height of the cyclotomic polynomial ${\Phi }_n$ is at least $(1-\epsilon )c_{\omega }{M}_n$, where $M_n={\prod }_{i=1}^{\omega -2}{p}_i^{2^{\omega -1-i}-1}$ and $c_{\omega }$ is a constant depending only on $\omega$; furthermore ${\lim }_{\omega \to \infty }{c}_{\omega }^{2^{-\omega }}\approx 0.71$. In our construction we can have $p_i>h(p_1{p}_2\dots p_{i-1})$ for all $i=1,2,\dots ,\omega$ and any function $h:{\mathbb{R}}_{+}\to {\mathbb{R}}_{+}$.

8 pages