The Strong Chowla-Milnor spaces and a conjecture of Gun, Murty and Rath

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 $\zeta (k)\mathrm{/}{\pi }^k$ 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 $d\ge 1$, is at least 2, assuming a conjecture of Gun, Murty and Rath.