Timeline for Asymptotic equivalence for functions with zeros

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May 19, 2012 at 11:16	comment	added	Kevin Smith		@ Brendan: Your suggestions don't allow the cases when $f$ and $g$ don't eventually vanish at the same places, and they both amount to $\|f/g\|\rightarrow 1$ if $g$ doesn't vanish eventually. My reference to the possible implications of RH is not related to the zeros of $f$ and $g$, it is to do with the behavior of $\zeta (2\rho)$. Although I think RH is indirectly related to the zeros of $f$ and $g$ through some old theorems of Polya and Turan.
May 18, 2012 at 13:18	comment	added	Brendan McKay		Maybe something like $f(x)=(1+o(1))g(x) + o(1)$ with suitably small $o(1)$ terms is what you need.
May 18, 2012 at 12:32	comment	added	Kevin Smith		In the case I have in mind they are not eventually zero at the same places, so perhaps extending the definition of asymptotic equivalence is too strong. Specifically, I am considering $L(x)$ and $M(x)$, so your suggestion would lead me to attempting to prove or disprove the statement $L(x)=(1+o(1))M(x)$. This is exactly the kind of thing I am getting at, as it seems it may not be the case if RH holds, yet the failure of RH appears to imply it.
May 18, 2012 at 11:57	history	answered	Brendan McKay	CC BY-SA 3.0