Über die Eulersche ϕ-funktion in quadratischen Zahlkörpern
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On Euler's ϕ-function in quadratic number fields
Abstract
Let ϕ(a) denote the Euler totient function in an arbitrary quadratic number fieldK and defineE K (x) andH K (x) as the error terms in the asymptotic formulae for\(\sum \varphi (\mathfrak{a})\) and\(\sum \varphi (\mathfrak{a})N(\mathfrak{a})^{ - 1} \), respectively, summation being extended over all ideals a with 1≤N(a)≤x. In this paper the asymptotic behaviour of ∑ n=1 N E K (n) and ∑ n=1 N H K (n)H K (itn) is studied. This generalizes results ofPillai andChowla [5] on the classical case.
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