$\mathfrak{d}_{\mathfrak{p}}^{-1}$ but nontrivial on ${\mathfrak{p}}^{-1}\mathfrak{d}_{\mathfrak{p}}^{-1}$. Here and later a subscript ${\mathfrak{p}}$ indicates completion with respect to the corresponding valuation $v_{\mathfrak{p}}$. If $\Omega:=\prod_{\mathfrak{p}}{\mathfrak{o}}_{\mathfrak{p}}^\times$ is the unique maximal compact subgroup of $\mathbb{A}_{\text{fin}}^{\times}$, then $\Omega\backslash\mathbb{A}_{\text{fin}}^{\times}$ is isomorphic in a natural way to the multiplicative group $I(K)$ of nonzero fractional ideals of $K$. We shall occasionally identify an idele in $\mathbb{A}_{\text{fin}}^{\times}$ with its image under the corresponding surjective homomorphism $\mathbb{A}_{\text{fin}}^{\times}\to I(K)$. This homomorphism also gives rise to a natural action of $\mathbb{A}_{\text{fin}}^{\times}$ on $I(K)$. We write $\sim$ for equivalence in the ideal class group $$C(K):=K^\times\backslash I(K)\cong
K^\times\Omega\backslash\mathbb{A}_{\text{fin}}^{\times}\cong
K^\times K_\infty^\times\Omega\backslash\mathbb{A}^{\times}.$$ We write $h:=\#C(K)$ for the class number of $K$. Let ${\mathcal{N}}: K \to {\mathbb{Q}}$ be the norm, which we extend to an ${\mathbb{R}}$-multilinear map $K_\infty
\to {\mathbb{R}}$; the norm of a fractional ideal $\mathfrak{m}\in I(K)$ will also be denoted by ${\mathcal{N}}\mathfrak{m}$. Note that the norm of an infinite idele $y\in K_\infty^\times$ is $|y|$, but the norm of the fractional ideal $(y)=y{\mathfrak{o}}\in I(K)$ corresponding to a finite idele $y\in\mathbb{A}_\text{fin}$ is $|y|^{-1}$.

We denote by $\mu({\mathfrak{a}})$, $\varphi({\mathfrak{a}})$ and $\tau({\mathfrak{a}})$ the obvious generalizations of the Möbius, the Euler, and the divisor functions to nonzero ideals ${\mathfrak{a}}\subseteq{\mathfrak{o}}$. We will often use the basic estimates $\#\{\mathfrak{m} \subseteq {\mathfrak{o}}\mid
{\mathcal{N}}\mathfrak{m} {\leqslant}x 