A note on \(q\)-analogues of Dirichlet \(L\)-functions (Q2800157)

Let \(N\) be a positive integer and \(k\) be an integer. Let \(\chi\) be a primitive Dirichlet character modulo \(N\) such that \(\chi(-1)=(-1)^k\). Then the authors prove the following result.NEWLINENEWLINEIf \(q\) is an algebraic number such that \(\mid q\mid <1\) then the number NEWLINE\[NEWLINEL_q(k,\chi)=\sum_{n=1}^\infty \left(\sum_{d\mid n} \chi (\frac nd) d^{k-1}\right)q^nNEWLINE\]NEWLINE is transcendental.NEWLINENEWLINEIf \(q=e^{2\pi i\tau}\) and \(j(\tau)\in \overline{\mathbb Q}\), where \( j(\tau)=1728\left(1-\frac{E_6(\tau)^2}{E_4(\tau)^3}\right)^{-1}\), NEWLINE\[NEWLINE\begin{aligned} E_4(\tau)=1+240\sum_{n=1}^\infty \sigma_3(n)q^n,\\ E_6(\tau)=1-504\sum_{n=1}^\infty \sigma_5(n)q^n,\end{aligned}NEWLINE\]NEWLINE \(\sigma_s(n)=\sum_{d\mid n} d^s\) for all positive integers \(s\) and if \(k=2\) we assume that \(N\not= 1\), then there exists a transcendental number \(\omega_\tau\) which depends only on \(\tau\) and is \(\overline{\mathbb Q}\)-linearly independent with \(\pi\) such that \((\frac {\pi}{\omega_\tau})^k(L(1-k,\chi)+2L_q(k,\chi))\) is algebraic where NEWLINE\[NEWLINEL(s,\chi)=\frac 1{(s-1)!}\lim_{q\to 1} (1-q)^sL_q(k,\chi)NEWLINE\]NEWLINE for all positive integers \(s\).