Linear independence of harmonic numbers over the field of algebraic numbers. II (Q780444)

For every complex non-negative integer \(r\) let us denote \(H_r=\int_0^1 \frac{1-x^r}{1-x}\,dx\). Let \(n\) be a positive integer. Let \(\mathbb J=\{ m_i; i=1,\dots ,n\}\) be a finite set of pairwise co-prime odd integers such that for every \(m\in\mathbb J\) there exist primes \(p_1\), \(p_2\) and \(a,b\in\mathbb N\) with \(m=p_1^ap_2^b\), \((a,\phi(p_2^b))=(b,\phi(p_1^a))=1\) and satisfying 1. \(p_1\equiv p_2\equiv 3\pmod 4\): \(p_1\) and \(p_2\) are semi-primitive roots \(\mod p_2^b\), \(p_1^a\), respectively or 2. \(p_1\), \(p_2\) are primitive roots \(\mod p_2^b\) and \(\mod p_1^a\), respectively. Set \(W_\mathbb J =\overline{\mathbb Q} -\text{span of}\ \{ H_1, H_{a_{j_i}/m_i};1\leq a_{j_i}, j_i\leq m_i-1, \ 1\leq i\leq n,\ m_i\in\mathbb J \}\). Then the authors prove that \[ \dim_{\overline{\mathbb Q}}W_\mathbb J = \sum_{i=1}^n\frac {\phi(m_i)}2+n+3. \] For Part I, see Ramanujan J. 51, No. 1, 43--66 (2020; Zbl 1441.11180).