Shintani's zeta function is not a finite sum of Euler products

2011/12/06 by Frank Thorne, Thorne, Frank
Mathematics · #Advanced Algebra and Geometry #Advanced Mathematical Identities #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT) #math.NT

paper · pdf · doi:10.48550/arxiv.1112.1397

10 pages (lightly further revised; to appear in Proc. AMS)

openalex publication_date 2011/12/06 · arxiv created 2012/11/09 · arxiv updated 2012/11/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We prove that the Shintani zeta function associated to the space of binary cubic forms cannot be written as a finite sum of Euler products. Our proof also extends to several closely related Dirichlet series. This answers a question of Wright in the negative.

Citations

• On the Davenport-Heilbronn theorems and second order terms

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