In this clearly and well written paper the authors first discuss and then apply the technique of Möbius inversion from analytic number theory to the Fourier series expansions of Bernoulli polynomials. (It is mentioned that the Möbius inversion of Fourier series is a special cases of a general theory that extends far beyond the case of Fourier series and which the authors trace back to an idea of E. Cesàro [Brioschi Ann. (2) 13, 339–351 (1885; JFM 17.0224.01)]. Other papers are cited.) One obtained result is that \(\cos(2\pi x)\) can be expanded using a (fixed) even \(2k\) degree Bernoulli polynomial evaluated at the fractional parts of \(nx\) for all positive integers \(n\). Up to a regularization constant depending on \(k\) the coefficients are given by the Möbius function \(\mu(n)\) divided by \(n^{2k}\). For \(\sin(2\pi x)\) one has to change to odd degree \(2k+1\). Similar results for Euler polynomials are deduced. As an illustrative example for this method the authors provide the expansion of the constructible number \(\cos(2\pi/17)\) using the Bernoulli polynomial \(B_2\) evaluated at the rationals \(n/17\) modulo \(1\). The evaluation of the Fourier expansions of Bernoulli polynomials and the corresponding Möbius inverse at rational arguments \(x = r / m\) is the second main topic of the paper. The periodicity modulo \(m\) introduced into the series in this way lead to sums \(\sum 1/n^k\) and \(\sum \mu(n)/n^k\), where the sums are extended over positive integers \(n\) congruent \(r\) modulo \(m\). In an elementary manner, using only linear algebra and an auxiliary result involving matrices defined by Bernoulli polynomials and the cosine function, expressions for certain combinations of such sums are evaluated explicitly. This approach is in the spirit of P. Codecà, R. Dvornicich and U. Zannier [J. Théor. Nombres Bordx. 10, No.1, 49–64 (1998; Zbl 0929.11003)], which studies the case \(k=1\). Finally, the authors show asymptotic properties concerning Bernoulli and Euler polynomials. The results are not new but the proofs are simpler and, in addition, the rate of convergence is investigated.