Arithmetic properties of series of certain classes (Q2862894)

Let \(\mathbb Z^+\) be the set of all positive integers, let \(M\in\mathbb Z^+\) and let \(\{p_n\}_{n=1}^\infty\) be the increasing sequence of all primes. For every \(m,i\in\mathbb Z^+\) and \(j\in\{ 1,\dots ,M\}\) let \(\gamma_{i,j,m}\in\mathbb Z^+\) such that \(\gamma_{i,j,m}\leq \gamma_{i,j,m+1}\) and NEWLINE\[NEWLINE\frac{K(\log m +\log p_m\sum_{s=1}^m\gamma_{s,j,m})}{\log p_j}< \gamma_{i,j,m+1},NEWLINE\]NEWLINE where \(K\) is a constant with \(K>2\) and \(\log x\) is the natural logarithm of \(x\). For every \(m\in\mathbb Z^+\) and \(j\in\{ 1,\dots ,M\}\) set \(b_{j,m}=\prod_{p_i<p_m}p_i^{\gamma_{i,j,m}}\). Then the author proves that for every prime \(p\) the numbers \(\alpha_j=\sum_{m=1}^\infty b_{j,m}\), \(j\in\{ 1,\dots ,M\}\) are algebraically independent over \(\mathbb Q_p\).