A partial result towards the Chowla--Milnor conjecture

The Chowla--Milnor conjecture predicts the linear independence of certain Hurwitz zeta values. In this paper, we prove that for any fixed integer $k \geqslant 2$, the dimension of the $\mathbb{Q}$-linear span of $\zeta(k,a/q)-(-1)^{k}\zeta(k,1-a/q)$ ($1 \leqslant a < q/2$, $\gcd(a,q)=1$) is at least $(c_k -o(1)) \cdot \log q$ as the positive integer $q \to +\infty$ for some constant $c_k>0$ depending only on $k$. It is well known that $\zeta(k,a/q)+(-1)^{k}\zeta(k,1-a/q) \in \overline{\mathbb{Q}}\pi^k$, but much less is known previously for $\zeta(k,a/q)-(-1)^{k}\zeta(k,1-a/q)$. Our proof is similar to those of Ball--Rivoal (2001) and Zudilin (2002) concerning the linear independence of Riemann zeta values. However, we use a new type of rational functions to construct linear forms.