A factor of integer polynomials with minimal integrals

Carlo Sanna

doi:10.5802/jtnb.994

Carlo Sanna ¹

¹ Università degli Studi di Torino Department of Mathematics Via Carlo Alberto 10 10123 Torino, Italy

Journal de théorie des nombres de Bordeaux, Volume 29 (2017) no. 2, pp. 637-646.

Abstract
Résumé

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

Pour chaque entier positif $N$ , soit $S_{N}$ l’ensemble des polynômes $P (x) \in ℤ [x]$ de degré inférieur à $N$ et d’intégrale positive non-nulle minimale sur $[0, 1]$ . Ces polynômes sont liés à la répartition des nombres premiers puisque, si l’on note $ψ$ la fonction de Tchebychev, on a $\int_{0}^{1} P (x) d x = exp (- ψ (N))$ . Nous démontrons que, pour tout nombre entier positif $N$ , il existe $P (x) \in S_{N}$ tel que le polynôme ${(x (1 - x))}^{⌊ N / 3 ⌋}$ divise $P (x)$ dans $ℤ [x]$ . Nous montrons en fait que l’exposant $⌊ N / 3 ⌋$ ne peut pas être amélioré. Ce résultat est analogue à celui obtenu par Aparicio concernant les polynômes de $ℤ [x]$ de norme $L^{\infty}$ non-nulle minimale sur $[0, 1]$ . En outre, il est en quelque sorte l’amélioration d’un résultat de Bazzanella, qui considérait $x^{⌊ N / 2 ⌋}$ et ${(1 - x)}^{⌊ N / 2 ⌋}$ au lieu de ${(x (1 - x))}^{⌊ N / 3 ⌋}$ .

Received: 2016-02-03
Revised: 2016-02-17
Accepted: 2016-02-26
Published online: 2017-06-12

DOI: 10.5802/jtnb.994

Classification: 11A41, 11C08, 11A63
Mots-clés : Integer polynomials, Chebyshev problem, prime numbers

Author's affiliations:

Carlo Sanna ¹

¹ Università degli Studi di Torino Department of Mathematics Via Carlo Alberto 10 10123 Torino, Italy

License:

CC-BY-ND 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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     title = {A factor of integer polynomials with minimal integrals},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
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Carlo Sanna. A factor of integer polynomials with minimal integrals. Journal de théorie des nombres de Bordeaux, Volume 29 (2017) no. 2, pp. 637-646. doi : 10.5802/jtnb.994. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.994/

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