A partial result towards the Chowla--Milnor conjecture

TL;DR

This paper establishes a lower bound on the dimension of the rational linear span of certain Hurwitz zeta differences, advancing understanding of the Chowla--Milnor conjecture through novel rational functions.

Contribution

It provides the first nontrivial lower bound on the dimension of these Hurwitz zeta value differences, using new rational functions in the proof.

Findings

The dimension grows at least proportionally to log q for fixed k.

The method extends techniques from Riemann zeta value independence proofs.

It introduces a new approach with rational functions to construct linear forms.

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 $\zeta(k,a/q)-(-1)^{k}\zeta(k,1-a/q)$ ($1 \leqslant a < q/2$, $\gcd(a,q)=1$) is at least $(c -o(1)) \cdot \log q$ as the positive integer $q \to +\infty$ for some absolute constant $c>0$. 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.