Another factor of integer polynomials with minimal integrals

Abstract

Let $N$ be a positive integer and let $S_N$ be the set of polynomials with integer coefficients, degree less than $N$, and minimal positive integral over $[0,1]$. D. Bazzanella initiated the study of $S_N$ because of its relation to the distribution of prime numbers. Indeed, it is possible to prove that ${\sum }_{p^m\le N}\log p=-\log {\int }_0^{1}P(x)\mathrm{d}x$ for every $P\in S_N$, where the sum runs over prime numbers $p$ and positive integers $m$ such that $p^m\le N$. For each real number $t$, let $\lfloor t\rfloor$ denote the maximal integer not exceeding $t$. The main result of this paper states that there exist infinitely many polynomials $P\in S_N$ such that $(x^3(1-x{)}^2{)}^{\lfloor N/6\rfloor }$ divides $P(x)$ in $\mathbb{Z}[x]$. This improves upon a similar result of Sanna, who proved the same claim but with the lower-degree polynomial $(x(1-x){)}^{\lfloor N/3\rfloor }$ in place of $(x^3(1-x{)}^2{)}^{\lfloor N/6\rfloor }$.