On the dimension of Chowla-Milnor space

In a recent work, Gun, Murty and Rath defined the Chowla-Milnor space and proved a non-trivial lower bound for these spaces. They also obtained a conditional improvement of this lower bound and noted that an unconditional improvement of their lower bound will lead to irrationality of $\zeta(k)/ \pi^k$ for odd positive integers $k>1$. In this paper, we give an alternate proof of their theorem about the conditional lower bound.