A non-commutative multiple Dirichlet power product and an application (Q1729760)

The author extends Möbius inversion formula to complex-valued functions of ideals of a Dedekind domain \(O\). For two \(k\)-variable ideal functions \(f\) and \(g\), the author defines their Dirichlet \(r\)-product by \[f*_rg(A_1,\ldots,A_k)=\sum_{B^r\mid A_i\ \text{for}\ 1\le i\le k}f(B^r,\ldots,B^r)g\left(\frac{A_1}{B^r},\ldots,\frac{A_k}{B^r}\right),\] where \(A_1,\ldots,A_k\) and \(B\) are ideals of \(O\). This product is not commutative but satisfies the Möbius inversion formula. As an application, the author deduces a formula for the number of lattice points \((A_1,\ldots,A_m)\) with norms \(N(A_1),\ldots, N(A_m)\) not exceeding \(x\) and \(P^r\supseteq\bigcup_{i=1}^m A_i\) for no prime ideal \(P\).