A partial result towards the Chowla--Milnor conjecture

2025/05/19 by Li Lai, Jia Li, Lai, Li +1
Mathematics · #11J72 (primary) #11M35 #33C20 (secondary) #Advanced Mathematical Identities #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2505.12687

openalex publication_date 2025/05/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 ℚ-linear span of ζ(k,a/q)-(-1)^kζ(k,1-a/q) (1 \leqslant a < q/2, gcd(a,q)=1) is at least (c -o(1)) ⋅ log q as the positive integer q → +∞ for some absolute constant c>0. It is well known that ζ(k,a/q)+(-1)^kζ(k,1-a/q) ∈ ℚπ^k, but much less is known previously for ζ(k,a/q)-(-1)^kζ(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.

Citations

• Nesterenko's criterion when the small linear forms oscillate

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