Zeros of the Dirichlet series of even zeta values

[2607.20758] Zeros of the Dirichlet series of even zeta values Skip to main content Search Submit Donate Log in Mathematics > Number Theory arXiv:2607.20758 (math) [Submitted on 22 Jul 2026 ( v1 ), last revised 22 Aug 2026 (this version, v2)] Title: Zeros of the Dirichlet series of even zeta values Authors: William D. Banks View PDF HTML (experimental) Abstract: We give a complete unconditional description of the zero set of the Dirichlet series $\mathbf{D}(s) = \sum_{n \ge 1} \zeta(2n)n^{-s}$, which continues meromorphically to $\mathbb{C}$ with a single simple pole at $s = 1$. The series possesses neither an Euler product nor a self-dual functional equation, and descriptions with this level of completeness are exceedingly rare for such series. The key input is an exact functional equation of Hecke type, obtained from the Lipschitz summation formula, which expresses $\mathbf{D}$ in the left half-plane as a gamma factor times a dual series over the complex logarithms of the perfect squares; Riemann's functional equation appears as a single column of the dual series. The zeros fall into four families. The half-plane $\sigma \ge \sigma_0 = 1.5001\ldots$ is zero-free, and $\mathbf{D}$ has a unique real zero $\rho_0 = 0.2004\ldots$. The zeros in the critical strip are perturbed $a$-points of $\zeta$ for values of $a$ near $-(\zeta(2)-1)$, and their counting function obeys a Riemann--von Mangoldt law. The remaining zeros form two complex-conjugate strings that recede into the left half-plane along explicit rays, are eventually simple, and satisfy an asymptotic with geometrically decaying error. The string geometry is governed by interference between the two smallest frequencies of the dual series, $2\log 2$ contributed by the entire part of $\mathbf{D}$ and $\pm 2\pi i$ contributed by $\zeta$. No hypothesis of Riemann type is assumed at any point. Comments: v2: Conjecture 4.3 of v1 is now Proposition 4.3, proved following an observation of Qiping Zhou; two references added. No other result is affected. 26 pages, 4 figures, 3 tables Subjects: Number Theory (math.NT) MSC classes: 11M41 (Primary), 11M06, 11M26 (Secondary) Report number: MPIM-Bonn-2026 Cite as: arXiv:2607.20758 [math.NT] (or arXiv:2607.20758v2 [math.NT] for this version) https://doi.org/10.48550/arXiv.2607.20758 Focus to learn more arXiv-issued DOI via DataCite Submission history From: William Banks [ view email ] [v1] Wed, 22 Jul 2026 22:19:52 UTC (91 KB) [v2] Sat, 22 Aug 2026 12:26:29 UTC (92 KB) Full-text links: Access Paper: View PDF HTML (experimental) TeX Source view license Current browse context: math.NT < prev | next > new | recent | 2026-07 Change to browse by: math References & Citations NASA ADS Google Scholar Semantic Scholar export BibTeX citation 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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