Linear independence of Hurwitz zeta values and a theorem of Baker-Birch-Wirsing over number fields (Q2919689)

Let \(x\in\mathbb R\) with \(0<x\leq 1\) and \(s\in\mathbb C\) with \(\mathrm{Re}(s)>1\). Define the Hurwitz zeta function as \(\zeta(s,x)=\sum_{n=0}^\infty \frac 1{(n+x)^s}\). Let \(\mathbb F\) be a number field and \(k\) and \(q\) be integers such that \(k,q>1\). Let \(V_k(q,\mathbb F)\) be the \(\mathbb F\)-linear space defined by \(V_k(q,\mathbb F)=\mathbb F-\text{span \;of} \;\{\zeta(k,a/q); 1\leq a<q, (a,q)=1\}\). The authors prove several upper and lower bounds for \(V_k(q,\mathbb F)\) under the different assumptions. In addition, for a periodic function \(f\) with period \(q\) let us define the \(L\)-series \(L(s,f)=\sum_{n=1}^\infty \frac {f(n)}{n^s}\). The authors generalize the Baker-Birch-Wirsing theorem as follows.NEWLINENEWLINETheorem 1: For an integer \(q>1\), let \(f\) be a periodic function with the period \(q\) taking values in a number field \(\mathbb F\). Further, suppose \(f(a)=0\) whenever \(1<(a,q)<q\). Also, let \(\mathbb K=\mathbb F\cap\mathbb Q(\zeta_q)\) and \(H=\mathrm{Gal}(\mathbb Q(\zeta_q)/\mathbb K)\subseteq (\mathbb Z/q\mathbb Z)^*\). Assume that the support of \(f\) in \(\mathbb Z/q\mathbb Z\) is contained in \(H\cup \{q\}\). Then \(L(1,f)=0\) if and only if \(f\equiv 0\).NEWLINENEWLINEFinally they also prove that Conjecture 1 holds if Polylog Conjecture holds where Conjecture 1 and Polylog Conjecture are defined as follows.NEWLINENEWLINEPolylog Conjecture: For an integer \(k\geq 2\) and complex number \(z\) with \(| z| \leq 1\) let us define \(\mathrm{Li}_k(z)=\sum_{n=1}^\infty \frac {z^n}{n^k}\). Assume that \(m\) be a positive integer. Suppose that \(\alpha_1,\dots ,\alpha_m\) are algebraic numbers with \(|\alpha_n|\leq 1\) for all \(n\in\{ 1,\dots ,m\}\) such that \(\mathrm{Li}_k(\alpha_1),\dots ,\mathrm{Li}_k(\alpha_m)\) are linearly independent over \(\mathbb Q\). Then they are linearly independent over the field of algebraic numbers.NEWLINENEWLINEConjecture 1: Let \(q>1\) be an integer and \(\mathbb F\) be a number field such that \(\mathbb F\cap \mathbb Q(\zeta_q)=\mathbb Q\). Then \(\dim_{\mathbb F} V_k(q,\mathbb F)=\phi(q)\) for all integers \(k>1\).