Search the site

Table of contents for this issue | Previous article | Next article
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
Article PDF
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 $(x(1-x))^{\lfloor N / 3 \rfloor }$.


[1] Emiliano Aparicio Bernardo, On the asymptotic structure of the polynomials of minimal diophantic deviation from zero, J. Approximation Theory 55 (1988), p. 270-278 Article |  MR 968933
[2] Danilo Bazzanella, A note on integer polynomials with small integrals, Acta Math. Hung. 141 (2013), p. 320-328 Article
[3] Danilo Bazzanella, A note on integers polynomials with small integrals II, Acta Math. Hung. 146 (2016), p. 71-81 Article
[4] Peter Borwein & Tamás Erdélyi, The integer Chebyshev problem, Math. Comput. 65 (1996), p. 661-681 Article
[5] Pafnutiĭ L’vovich Chebyshev, Collected works. Vol. I, Akad. Nauk SSSR, 1944
[6] Alan Jeffrey & Hui-Hui Dai, Handbook of mathematical formulas and integrals, Elsevier/Academic Press, 2008
[7] Ernst Eduard Kummer, Über die Ergänzungssätze zu den allgemeinen Reciprocitätsgesetzen, J. Reine Angew. Math. 44 (1852), p. 93-146 Article
[8] Hugh L. Montgomery, Ten lectures on the interface between analytic number theory and harmonic analysis, Regional Conference Series in Mathematics 84, American Mathematical Society, 1994
[9] Mohan Nair, A new method in elementary prime number theory, J. Lond. Math. Soc. 25 (1982), p. 385-391 Article
[10] Mohan Nair, On Chebyshev-type inequalities for primes, Am. Math. Mon. 89 (1982), p. 126-129 Article
[11] Igor E. Pritsker, Small polynomials with integer coefficients, J. Anal. Math. 96 (2005), p. 151-190 Article