A factor of integer polynomials with minimal integrals

作者：Sanna Carlo^*

来源：Journal de Theorie des Nombres de Bordeaux, 2017, 29(2): 637-646.

DOI：10.5802/jtnb.994

• Integer polynomials
• Chebyshev problem
• prime numbers

For each positive integer N, let S-N be the set of all polynomials P(x) is an element of 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 integral (1)(0) P(x) dx = exp(-psi(N)), where psi is the second Chebyshev function. We prove that for any positive integer N there exists P(x) is an element of 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-infinity 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]).

• 出版日期2017