A generalisation of an identity of Lehmer (Q2809364)

Let \(\Omega\) be a finite subset of primes and NEWLINE\[NEWLINE P_\Omega=\prod_{p\in\Omega}p, \qquad \delta_\Omega=\prod_{p\in\Omega}1-1/p,NEWLINE\]NEWLINE where by convention an empty product is \(1\). Then the authors define the generalized Euler-Briggs constant \(\gamma(\Omega,a,q)\) by NEWLINE\[NEWLINE\gamma(\Omega,a,q)=\lim_{x\rightarrow \infty} \sum_{\substack{ n\leq x \\ (n,P_\Omega)=1 \\ n\equiv a \mod q}} \frac 1n-\frac{\delta_{\Omega}}{q}\log x.NEWLINE\]NEWLINE Let us note that \(\gamma(\emptyset,1,1)=\gamma\) is the usual Euler constant and \(\gamma(\emptyset,a,q)=\gamma(a,q)\) is the Euler-Briggs constant.NEWLINENEWLINEThe main result of the paper under review is a generalization of a result of \textit{D. H. Lehmer} [Acta Arith. 27, 125--142 (1975; Zbl 0302.12003)], who showed that NEWLINE\[NEWLINEq\gamma(a,q)-\gamma=\sum_{\substack{ \zeta_q \in \mu_q \\ \zeta_q\neq 1}} \zeta_q^{-a} \log (1-\zeta_q),NEWLINE\]NEWLINE where \(\mu_q\) is the set of \(q\)-th roots of unity. In particular, they show that NEWLINE\[NEWLINE\gamma(\Omega,a,q)-\delta_{\Omega} \frac{\gamma}a= \frac{\delta_\Omega}q \sum_{p\in \Omega} \frac{\log p}{p-1}- \sum_{\Omega'\subseteq \Omega}\frac{(-1)^{|\Omega'|}}{qP_{\Omega'}} \sum_{\substack{ \zeta_q \in \mu_q \\ \zeta_q\neq 1}} \zeta_q^{-a} \log \left(1-\zeta_q^{P_{\Omega'}}\right). NEWLINE\]NEWLINE Using Baker's theory of linear forms in logarithms they obtain as a corollary that the numbers \(\gamma(\Omega,a,q)-\delta_{\Omega} \frac{\gamma}a\) are transcendent, if \(q>1\) and \((a,q)=1\). Moreover, they can show that at least one of the numbers \(\gamma(\Omega,a,q)\) and \(\gamma(\Omega',a,q')\) is transcendental, provided that \((qq',P_\Omega P_{\Omega'})=1\), \((q,q')=1\) and \((a,q)=(a,q')=1\). Indeed, they prove an even more general statement.