The global field Euler function
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RESEARCH
The global field Euler function Santiago Arango-Piñeros1* * Correspondence:
[email protected] Department of Mathematics, Emory University, Atlanta, GA 30322, USA Full list of author information is available at the end of the article 1
and Juan Diego Rojas2
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 Erdos–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.
Contents 1 Introduction . . . . . . . . . . . . . . . . 1.1 Euler’s function . . . . . . . . . . . . 1.2 Global fields . . . . . . . . . . . . . . 1.3 Results . . . . . . . . . . . . . . . . . 2 Number fields . . . . . . . . . . . . . . . 2.1 Dedekind’s zeta function . . . . . . 2.2 The Euler function of a number field 3 Curves over finite fields . . . . . . . . . . 3.1 Weil divisors on a curve . . . . . . . 3.2 Weil’s zeta function . . . . . . . . . 3.3 The function field Euler function . . 4 Applications . . . . . . . . . . . . . . . . 4.1 Mean value of global field totients . 4.2 Mertens’ mean value . . . . . . . . . References . . . . . . . . . . . . . . . . . . .
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1 Introduction 1.1 Euler’s function
The classical arithmetical function of Euler, denoted by ϕ(n), is defined in group theoretic terms as the order of the group of units modulo n. The Chinese remainder theorem implies that ϕ(n) is a multiplicative arithmetical function, and this means that for every pair of relatively prime positive integers a and b, we have ϕ(ab) = ϕ(a)ϕ(b). Since for every prime number p and every positive integer r, the ring Z /pr Z is local with maximal ideal p Z /pr Z, and the units modulo pr correspond to the complement of p Z /pr Z inside Z /pr Z. This implies ϕ(pr ) = pr (1 − 1/p). The unique factorization property of the integers yields the
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S. Arango-Piñeros,
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