A partial result towards the Chowla--Milnor conjecture

[2505.12687] A partial result towards the Chowla--Milnor conjecture Skip to main content Search Submit Donate Log in Search arXiv Press Enter to search · Advanced search Mathematics > Number Theory arXiv:2505.12687 (math) [Submitted on 19 May 2025 ( v1 ), last revised 9 Apr 2026 (this version, v2)] Title: A partial result towards the Chowla--Milnor conjecture Authors: Li Lai , Jia Li View a PDF of the paper titled A partial result towards the Chowla--Milnor conjecture, by Li Lai and 1 other authors View PDF HTML (experimental) 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. Comments: 39 pages, 7 figures, v2 improved the constant by using a refined version of Nesterenko's linear independence criterion Subjects: Number Theory (math.NT) MSC classes: 11J72 (primary), 11M35, 33C20 (secondary) Cite as: arXiv:2505.12687 [math.NT] (or arXiv:2505.12687v2 [math.NT] for this version) https://doi.org/10.48550/arXiv.2505.12687 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Li Lai [ view email ] [v1] Mon, 19 May 2025 04:17:27 UTC (26 KB) [v2] Thu, 9 Apr 2026 21:10:02 UTC (26 KB) Full-text links: Access Paper: View a PDF of the paper titled A partial result towards the Chowla--Milnor conjecture, by Li Lai and 1 other authors View PDF HTML (experimental) TeX Source view license Current browse context: math.NT < prev | next > new | recent | 2025-05 Change to browse by: math References & Citations NASA ADS Google Scholar Semantic Scholar export BibTeX citation Loading... BibTeX formatted citation × loading... Data provided by: Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer ( What is the Explorer? ) Connected Papers Toggle Connected Papers ( What is Connected Papers? ) Litmaps Toggle Litmaps ( What is Litmaps? ) scite.ai Toggle scite Smart Citations ( What are Smart Citations? ) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv ( What is alphaXiv? ) Links to Code Toggle CatalyzeX Code Finder for Papers ( What is CatalyzeX? ) DagsHub Toggle DagsHub ( What is DagsHub? ) GotitPub Toggle Gotit.pub ( What is GotitPub? ) Huggingface Toggle Hugging Face ( What is Huggingface? ) ScienceCast Toggle ScienceCast ( What is ScienceCast? ) Demos Demos Replicate Toggle Replicate ( What is Replicate? ) Spaces Toggle Hugging Face Spaces ( What is Spaces? ) Spaces Toggle TXYZ.AI ( What is TXYZ.AI? ) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower ( What are Influence Flowers? ) Core recommender toggle CORE Recommender ( What is CORE? ) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs . Which authors of this paper are endorsers? | Disable MathJax ( What is MathJax? ) We gratefully acknowledge support from our major funders , member institutions , , and all contributors. About · Help · Contact · Subscribe · Copyright · Privacy · Accessibility · Operational Status (opens in new tab) Major funding support from

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