  In a recent work, Gun, Murty and Rath formulated the Strong Chowla-Milnor
conjecture and defined the Strong Chowla-Milnor space. In this paper, we prove
a non-trivial lower bound for the dimension of these spaces. We also obtain a
conditional improvement of this lower bound and noted that an unconditional
improvement of this lower bound will lead to irrationality of both $\zeta(k)$
and $\zeta(k)/ \pi^k$ for all odd positive integers $k>1$. Following Gun, Murty
and Rath, we define generalized Zagier spaces $V_p(K)$ for multiple zeta values
over a number field $K$. We prove that the dimension of $V_{4d+2}(K)$ for
$d\geq 1$, is at least 2, assuming a conjecture of Gun, Murty and Rath.
