  Using a remainder theorem for valuations of a field, we give a new
perspective on the norm function of a global field. We define the Euler totient
function of a global field and recover the essential analytical properties of
the classical arithmetical function, namely the product formula. In addition,
we prove the holomorphicity of the associated zeta function. As an application,
we recover the analog of the mean value theorem of Erd\H{o}s, Dressler, and
Bateman via the Weiner-Ikehara theorem.
