Cycles, Eulerian digraphs and the Schönemann-Gauss theorem (Q2156723)

In the paper, the author mainly proves the following generalization of Fermat's little theorem: For \(d\in\{1,2,3,\ldots\}\) let \(P_d(x)=x^r+a_{r-1}^{(d)}x^{r-1}+\cdots+a_0^{(d)}\) be the polynomial with integer coefficients such that the zeros of \(P_d\) are just the \(d\)-th power of the zeros of \(P_1\). Then for any positive integer \(n\) and \(j\in\{0,1,\ldots,r-1\}\), \(\sum_{d\mid n}a_j^{(d)}\mu\big(\frac nd\big)\equiv 0\pmod n\), where \(\mu (n)\) is the Möbius function. This result was implicitly given by \textit{M. Mazur} and \textit{B. V. Petrenko} [Jpn. J. Math. (3) 5, No. 2, 183--189 (2010; Zbl 1234.05016)].