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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 $s=\sigma+it$

(https://terrytao.wordpress.com/2014/12/09/254a-notes-2-complex-analytic-multiplicative-number-theory/#zzag (49) )

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 :)

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    $\begingroup$ 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. $\endgroup$ – Ian Oct 2 '18 at 14:09
  • $\begingroup$ Ah yes, that could also be the case... $\endgroup$ – C. Brendel Oct 2 '18 at 14:12
  • $\begingroup$ It also seems quite plausible :) I think that might be it. Still I'm going to wait for certainty! Thanks! $\endgroup$ – C. Brendel Oct 2 '18 at 14:27

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