A partial result towards the Chowla--Milnor conjecture
Li Lai, Jia Li
TL;DR
This work studies the arithmetic independence of Hurwitz zeta values in the odd Chowla–Milnor subspace $V_k^{-}(q)$ for fixed $k\ge 2$ as $q\to\infty$. The authors construct novel rational functions $R_n(t)$ to form linear forms $S_n$ in $1$ and elements of $V_k^{-}(q)$, and represent these forms via complex contour integrals. A careful saddle-point analysis identifies dominant saddle points and yields explicit exponential decay for $S_n$, enabling the use of Nesterenko’s linear independence criterion to bound the dimension of $V_k^{-}(q)$ from below. Consequently, they prove $\,\dim_{ mathbb{Q}} V_k^{-}(q) \ge \left(\frac{1}{k+\log 2}-o(1)\right) \cdot \log q$ as $q\to\infty$, offering meaningful partial evidence toward the Chowla–Milnor conjecture and advancing the understanding of the odd part of Hurwitz zeta values.
Abstract
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 $ζ(k,a/q)-(-1)^{k}ζ(k,1-a/q)$ ($1 \leqslant a < q/2$, $\gcd(a,q)=1$) 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 $ζ(k,a/q)+(-1)^{k}ζ(k,1-a/q) \in \overline{\mathbb{Q}}π^k$, but much less is known previously for $ζ(k,a/q)-(-1)^{k}ζ(k,1-a/q)$. 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.