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.

中文翻译：

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全局场欧拉函数

我们定义一个全局场的欧拉函数，并恢复经典算术函数的基本性质。此外，我们证明了相关zeta函数的全纯性。作为应用，我们恢复了Mertens和Erdős–Dressler–Bateman均值定理的类似物。博览会针对算术统计领域的非专家，旨在为深入了解算术功能在算术拓扑结构中的其他情况提供一般性信息。