  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.
