asymptotic of Riemann xi function on the critical line

Question

given the Xi function on the critical line $ \xi ( 1/2+iz) $

then for big $ z \to \infty $

would be the following asymptotic be valid ?

$$ \xi( 1/2+iz) \sim g(z) $$

where $ g(x)$ is a real function with real roots and the roots of 'g' satify the following asymptotic condition

$ f(c_n ) =0 $ , $ c_{n} = \frac{2\pi n}{W(ne^{-1})}$

here $ W(z)$ is the Lambert function

for the case of Bessel function and Airy function in Quantum (semiclassical) mechanics this seems to work

This seems to be related to [this previous question of yours](http://math.stackexchange.com/q/714679/5531). I hoped my and Raymond's answers there would convince you that the answer to this is "no". — 2017-01-12
$f(x) \sim g(x)$ as $x \to \infty$ means $\lim \frac{f(x)}{g(x)}= 1$. So clearly no if they have infinitely many zeros and not at the same $x$. Now maybe you meant $\sup_{y < x} f(y) \sim \sup_{y < x} g(y)$ @AntonioVargas — 2017-04-06

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