Transcendence of generalised Euler-Kronecker constants (Q6564337)

Let \(n\) be a positive integer, let \(\mathbb K\) be a number field of degree \(n\) and let \(\mathbf O_{\mathbb K}\) denote its ring of integers. For \(s\) with \(\Re (s)>1\) set \[\zeta_{\mathbb K}(s)=\sum_{(0)\not=J\subseteq \mathbf O_{\mathbb K}} \frac 1{N(J)^s}= \frac {r_{\mathbb K}}{s-1}+c_{\mathbb K}+O(s-1) \] where \(N(J)\) is a norm of the ideal \(J\) and \(r_{\mathbb K}\) is residue of \(\zeta_{\mathbb K}\) at \(s=1\). Set also \(\gamma_{\mathbb K}=\frac {c_{\mathbb K}}{r_{\mathbb K}}\). Let \(P_{\mathbb K}\) denote the set of non-zero prime ideals \(p\) of \(\mathbf O_{\mathbb K}\). Let \(A=\{\Omega_i; i\in I\}\) be a family of non-empty subsets of \(P_{\mathbb K}\) such that for every \(i\in I\) we have \(\sum_{p\in\Omega_i} \frac {\log N(p)}{N(p)-1}<\infty\). For every \(\Omega\in A\) and real \(x\) set \(N_\Omega=\{ p\cap\mathbb Z; p\in\Omega\}\), \(\Omega(x)=\{ p\in\Omega ; N(p)\leq x\}\), \(P(\Omega(x))=\prod_{p\in\Omega(x)}p\), \(\delta_{\mathbb K}(\Omega)=\prod_{p\in\Omega} (1-\frac 1{N(p)})\) and \(\gamma_{\mathbb K}(\Omega)=\lim_{x\to\infty} (\frac 1{r_{\mathbb K}}\sum_{0\not= a\subset\mathbf O_{\mathbb K}, N(a)\leq x, (a,P(\Omega(x)))=1} \frac 1{N(a)}- (\log x)\prod_{p\in\Omega(x)}(1-\frac 1{N(a)}))\). Assume that for every \(i,j\in I\), \(i\not=j\) the set \(N_{\Omega_i}\setminus N_{\Omega_j}\) is non-empty. Then at most one number from the set \(\{ \frac {\gamma_{\mathbb K}(\Omega_i)}{\delta_{\mathbb K}(\Omega_i)}; i\in I\}\) is algebraic.