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Carlo Sanna
A factor of integer polynomials with minimal integrals
Journal de théorie des nombres de Bordeaux, 29 no. 2 (2017), p. 637-646, doi: 10.5802/jtnb.994
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Class. Math.: 11A41, 11C08, 11A63
Keywords: Integer polynomials, Chebyshev problem, prime numbers

Résumé - Abstract

For each positive integer $N$, let $S_N$ be the set of all polynomials $P(x) \in \mathbb{Z}[x]$ with degree less than $N$ and minimal positive integral over $[0,1]$. These polynomials are related to the distribution of prime numbers since $\int _0^1 P(x) \,\mathrm{d}x = \exp (-\psi (N))$, where $\psi$ is the second Chebyshev function. We prove that for any positive integer $N$ there exists $P(x) \in S_N$ such that $(x(1-x))^{\lfloor N / 3 \rfloor }$ divides $P(x)$ in $\mathbb{Z}[x]$. In fact, we show that the exponent $\lfloor N / 3 \rfloor$ cannot be improved. This result is analog to a previous of Aparicio concerning polynomials in $\mathbb{Z}[x]$ with minimal positive $L^\infty$ norm on $[0,1]$. Also, it is in some way a strengthening of a result of Bazzanella, who considered $x^{\lfloor N / 2 \rfloor }$ and $(1-x)^{\lfloor N / 2 \rfloor }$ instead of $(x(1-x))^{\lfloor N / 3 \rfloor }$.

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