Rationality of the Euler–Mascheroni constant

Determine whether the Euler–Mascheroni constant γ, defined as the limit of the sequence b_n = ∑_{m=1}^{n} 1/m − log(n), is rational or irrational.

Background

In Section 3.2, while discussing examples of task types related to sequences and Cauchy convergence, the authors note that the sequence b_n = a_n − log(n), where a_n = 1 + 1/1 + 1/2 + … + 1/n, is Cauchy and hence convergent.

They then remark in a footnote that the limit of this sequence is the Euler–Mascheroni constant γ and explicitly state that it is still unknown whether γ is rational or irrational.