On a problem regarding coefficients of cyclotomic polynomials (Q1035583)

Let \[ \Phi_n(x) = \prod_{(j,n) = 1} ( x - \zeta_n^j) = \sum_{k=0}^{\varphi(n)} a_n(k) x^k \] be the \(n\)-th cyclotomic polynomial. Put \(a_n(k) = 0\) for \(k > \varphi(n)\). Let \[ M(k) = \lim_{x \rightarrow \infty} {1\over x}\sum_{n \leq x} a_n(k) \] be the average value of the \(k\)-th cyclotomic coefficient. It is shown that \[ f_k:={\pi^2\over 6}M(k)k \prod_{q \leq k,~q\text{~prime}}(q+1) \] is an integer for all \(k\). Earlier the reviewer has shown that \(2f_k\) is an integer, and Gallot on basis of numerical computations conjectured that \(f_k\) is always an integer [\textit{Y. Gallot}, the reviewer and \textit{H. Hommersom}, Value distribution of cyclotomic polynomial coefficients, \url{arXiv:0803.2483}]. The main result of the paper is that for every fixed natural integer \(N\), \(f_k\) is divisible by \(N\) for every \(k\) sufficiently large.