I recently stumbled upon some Landau notation I don't quite understand... It's the normal big-O notation with one or more indices, like: $$\log{\vert{L(s,\chi)}\vert}\le\mathcal{O}_\epsilon(\log{q(2+\vert{t}\vert)}$$ for some constants $0<\epsilon<\sigma$ with $s=\sigma+it$
Or in another case:
$$\frac{f'(s)}{f(s)}=\sum_{p:\vert{p-s_0}\vert\le c_1r}\frac{1}{s-p}+\mathcal{O}_{c_1,\,c_2}(\frac{\log{M}}{r})$$ for some constants $0<c_1<c_2<\sigma$ and some more other conditions on $M$ and $f$.
(https://terrytao.wordpress.com/2014/12/05/245a-supplement-2-a-little-bit-of-complex-and-fourier-analysis/ Theorem 21)
Does anybody know what this means? I have the suspicion that the author uses $\mathcal{O}_{c_1,\,c_2}(\frac{\log{M}}{r})$ as an placeholder for a function $g$ dependent on $c_1$ and $c_2$ bounded by $C\frac{\log{M}}{r}$ for some constant $C>0$. Probably with the intention not only to give a bound but also to make a statement about $g$ for which it ultimately is an placeholder. Vice versa $\mathcal{O}_\epsilon(\log{q(2+\vert{t}\vert})$should be a placeholder for some function $g(\epsilon)$.
Regards! Cedric :)