Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 137 / 1 / a / Solution

The Fourier expansion of a normalized Eisenstein series isThusThe Bernoulli number $B_k$ is rational, so every coefficient is rational. Equivalently, the standard Fourier calculation expresses the coefficient as a rational multiple of $(2\pi i{)}^k/\zeta (k)$, which is rational by the given fact that $\zeta (k)/{\pi }^k\in \mathbb{Q}$.

The supplied values give the familiar expansionsSince every divisor sum is an integer, all their coefficients are integers.