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I am feeling a bit slow today. In Analytic Number Theory it is usual to express asymptotic bounds by specifying the relation of the constant to a specific variable, i.e.

$\log n \ll_\epsilon n^\epsilon$

which means that $\log n \leqslant C_\epsilon n^\epsilon$ for sufficiently large $n$, where the constant $C_\epsilon$ depends only on the constant $\epsilon$.

Could someone explain what are the benefits of this versus just using the usual $\ll$.

I understand that $f \ll_\epsilon g \Rightarrow f \ll g$? Is it equivalent or is it a stronger statement?


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I think it's just a matter of clarity. Especially when there are a lot of parameters involved, it's nice to know which parameter a constant depends on. – froggie May 5 '12 at 20:00
I guess you are right and I think the discussion in this question answers my question pretty much. – yfyf May 6 '12 at 18:05

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