# Strange big-O notation?

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 $$0 and some more other conditions on $$M$$ and $$f$$.

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 :)

• I'd guess that $f(x,y)=O_y(g(x))$ means that $f(x,y)=O(g(x))$ for all fixed $y$, but the constant involved might depend on $y$. This is not exactly what you guessed. – Ian Oct 2 '18 at 14:09
• Ah yes, that could also be the case... – C. Brendel Oct 2 '18 at 14:12
• It also seems quite plausible :) I think that might be it. Still I'm going to wait for certainty! Thanks! – C. Brendel Oct 2 '18 at 14:27