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 $k⩾2$, the dimension of the $\mathbb{Q}$-linear span of $\zeta (k,a\mathrm{/}q)-(-1{)}^k\zeta (k,1-a\mathrm{/}q)$ ($1⩽a<q\mathrm{/}2$, $\gcd (a,q)=1$) is at least $(c-o(1))\cdot \log q$ as the positive integer $q\to +\mathrm{\infty }$ for some absolute constant $c>0$. It is well known that $\zeta (k,a\mathrm{/}q)+(-1{)}^k\zeta (k,1-a\mathrm{/}q)\in \bar{\mathbb{Q}}{\pi }^k$, but much less is known previously for $\zeta (k,a\mathrm{/}q)-(-1{)}^k\zeta (k,1-a\mathrm{/}q)$. 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.