Correction to: Aequat. Math. https://doi.org/10.1007/s00010-023-00967-w

CORRECTED FORMULA (1.10):

$$\begin{aligned} \mathcal {B}_k(z)&= -\frac{1}{\pi }\sum _{p/q\in \mathbb {Q}} \Bigg \{-(q'z-p')^k\left[ \text {Li}_2\left( \frac{p'-q'z}{qz-p}\right) - \text {Li}_2\left( -\frac{q'}{q}\right) \right] \\&\quad + (p''-q''z)^k\left[ \text {Li}_2\left( \frac{p''-q''z}{qz-p}\right) -\text {Li}_2\left( -\frac{q''}{q}\right) \right] \\&\quad + \sum _{n=1}^k \frac{1}{nq^n}\Bigg [-(q'z-p')^{k-n}\, \Bigg (\log (1+\frac{q'}{q})-\sum _{i=1}^{n-1}\frac{1}{i}\left( \frac{1}{(1+q'/q)^i}-1\right) \Bigg ) \\&\quad +(p''-q''z)^{k-n}\,\Bigg (\log (1+\frac{q''}{q})-\sum _{i=1}^{n-1}\frac{1}{i}\left( \frac{1}{(1+q''/q)^i}-1\right) \Bigg )\Bigg ] \Bigg \}, \end{aligned}$$

CORRECTED FORMULA (1.11):

$$\begin{aligned} \mathcal {W}(z)&= -\frac{1}{\pi }\sum _{p/q\in \mathbb {Q}}\Bigg \{(q'z-p')\left[ \text {Li}_2\left( \frac{p'-q'z}{qz-p}\right) -\text {Li}_2\left( -\frac{q'}{q}\right) \right] \\&\quad +(p''-q''z)\left[ \text {Li}_2\left( \frac{p''-q''z}{qz-p}\right) -\text {Li}_2\left( -\frac{q''}{q}\right) \right] \\&\quad +\frac{1}{q}\log \left( \frac{(q+q')(q+q'')}{q^2}\right) \Bigg \}\,. \end{aligned}$$