On linear independence of Dirichlet \(L\) values (Q2112776)

For a Dirichlet character \(\chi\) modulo \(q>1\) and \(s\in\mathbb C\) consider the \(L\)-Dirichlet function \(L(s,\chi)=\sum_{n=1}^\infty\frac{\chi(n)}{n^s}\). For any non-zero natural number \(q\) denote \(\zeta_q\) the primitive \(q\)-th root of unity, \(\chi_0\) the trivial character modulo \(q\) and \(\mathbb Q(\zeta_q)^+\) the maximal real subfield of \(\mathbb Q(\zeta_q)\). Then the author proves that if for \(1\leq j\leq l\), let \(q_j\) be pairwise co-prime natural numbers and \(l\geq 1\) be an integer. If \(\mathbb K\) be a number field such that \(\mathbb K(\zeta_{\phi(q_1)\dots \phi(q_l)})\cap\mathbb Q(\zeta_{q_1\dots q_L})^+=\mathbb Q\) then the numbers in the set \(\bigcup_{j=1}^l \{ L(2k+1,\chi)\mid \chi\bmod q_j, \ \chi(-1)=-1\}\) are linearly independent over \(\mathbb K(\zeta_{\phi(q_1)\dots \phi(q_l)})\) and also the numbers in the set \(\{ \zeta(2k)\}\cup \bigcup_{j=1}^l \{ L(2k,\chi)\mid \chi\bmod q_j, \ \chi(-1)=1, \ \chi\not= \chi_0 \}\) are linearly independent over \(\mathbb K(\zeta_{\phi(q_1)\dots \phi(q_l)})\). Then the author extends these results for the case of Chowla-Milnor vector spaces and the general Chowla-Milnor spaces in the spirit of dimensions.