A note on integer polynomials with small integrals. II (Q2400093)

For given \(N\geq2\) let \[ R_N=\{P\in Z[X]:\;\deg P<N, \int_0^1P(x) \,dx=0\}, \] \[ S_N=\{P\in Z[X]:\;\deg P<N, \int_0^1P(x) \,dx=1/d_N\}, \] where \(d_N=\mathrm{LCM}\{n\leq N\}\). Polynomials lying in \(S_N\) played some role in proofs of Chebyshev bounds for \(\pi(x)\), the number of primes \(p\leq x\). In the first part [Acta Math. Hung. 141, No. 4, 320--328 (2013; Zbl 1291.11062)] the author studied zeros of elements of \(S_N\). Now he constructs a basis of the \(Z\)-module \(R_N\) (Theorem 1), provides some information about the structure of \(S_N\) (Theorems 2--4), and disproves a conjecture stated in the first part.