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The global field Euler function

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Abstract

We define the Euler function of a global field and recover the fundamental properties of the classical arithmetical function. In addition, we prove the holomorphicity of the associated zeta function. As an application, we recover analogs of the mean value theorems of Mertens and Erdős–Dressler–Bateman. The exposition is aimed at non-experts in arithmetic statistics, with the intention of providing insight toward the generalization of arithmetical functions to other contexts within arithmetic topology.

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Notes

  1. one-dimensional separated scheme of finite type.

  2. We adopt this notation for the real and imaginary part of a complex variable s throughout the manuscript.

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Acknowledgements

We thank Guillermo Mantilla-Soler for very helpful discussions and suggestions.

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Authors and Affiliations

  1. Department of Mathematics, Emory University, Atlanta, GA, 30322, USA

    Santiago Arango-Piñeros

  2. Departmento de Matemáticas, Universidad de Los Andes, Bogotá, Colombia

    Juan Diego Rojas

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  1. Santiago Arango-Piñeros
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  2. Juan Diego Rojas
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Correspondence to Santiago Arango-Piñeros.

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Arango-Piñeros, S., Rojas, J.D. The global field Euler function. Res Math Sci 7, 19 (2020). https://doi.org/10.1007/s40687-020-00218-3

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  • DOI: https://doi.org/10.1007/s40687-020-00218-3

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