Sister Beiter conjecture

Short description: Conjecture on the coefficients of cyclotomic polynomials

In mathematics, the Sister Beiter conjecture is a conjecture about the size of coefficients of ternary cyclotomic polynomials (i.e. where the index is the product of three prime numbers). It is named after Marion Beiter, a Catholic nun who first proposed it in 1968.

Background

For $n\in {\mathbb{\mathbb{N}}}_{>0}$ the maximal coefficient (in absolute value) of the cyclotomic polynomial ${\mathrm{\Phi }}_n(x)$ is denoted by $A(n)$.

Let $3\le p\le q\le r$ be three prime numbers. In this case the cyclotomic polynomial ${\mathrm{\Phi }}_{pqr}(x)$ is called ternary. In 1895, A. S. Bang proved that $A(pqr)\le p-1$. This implies the existence of $M(p):=\max {}_{p\le q\le r\text{ prime}}A(pqr)$ such that $1\le M(p)\le p-1$.

Statement

Sister Beiter conjectured in 1968 that $M(p)\le \frac{p+1}{2}$. This was later disproved, but a corrected Sister Beiter conjecture was put forward as $M(p)\le \frac{2}{3}p$.

Status

A preprint from 2023 explains the history in detail and claims to prove this corrected conjecture. Explicitly it claims to prove $$M(p)\le \frac{2}{3}p\text{ and }\lim {}_{p\to \mathrm{\infty }}\frac{M(p)}{p}=\frac{2}{3}.$$