A factor of integer polynomials with minimal integrals

For each positive integer N, let SN be the set of all polynomials P(x)∈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 ∫10P(x)dx=exp(−ψ(N)), where ψ is the second Chebyshev function. We prove that for any positive integer N there exists P(x)∈SN 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∞ 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⌋.