On the radical of linear forms over the ring of arithmetical functions (Q2860988)

Let \(R\) be the ring of arithmetical functions in \(r\) variables with values in a field of zero characteristic with multiplication being a generalization of Dirichlet convolution. The authors showed earlier that \(R\) is a unique factorization domain and associated with every \(r\)-tuple \(t=(t_1,\dots,t_r)\) of positive \(\mathbb Q\)-linearly independent real numbers an absolute value \(|x|_t\) on \(R\) [ibid. 21, 29--39 (2008; Zbl 1141.11312)]. Now they determine an upper bound for the value of \(|\cdot|_t\) at the product of irreducible factors of the linear form \(\sum_{j=1}^mc_jx_j\) (with relatively prime non-zero \(c_j\in R\)) under the condition that the values \(|c_jx_j|_t\) are distinct.