Let $\zeta^{-1}(x)$ be the functional inverse of the Riemann zeta function so that $$\zeta(s) = x \implies \zeta^{-1}(x) = s$$
The Riemann zeta function is injective in $1 < s < \infty$ where $1 < \zeta(s) < \infty$. Hence in this interval $\zeta(s) = x$ has exactly one solution so we can ask for an asymptotic expansion of $\zeta^{-1}(x)$?
Question: What is known about the function $\zeta^{-1}(x)$ as $x \to \infty$? Is there any asymptotic expansion in literature or any other information.
Examples of things I am looking for: @GEdgar's comment suggests that the question may not be clear to some hence I am giving some examples.
I was working with inverse of the zeta function in real $x > 1$. For every $x > 1$ we can show using the Stieltjes series expansion of the Riemann zeta function about $s = 1$ that there exists a constant $0 < c_x < 1-\gamma$ such that $\zeta(1 + \frac{1}{x-1+c_x}) = x$ hence a first approximation is $\zeta^{-1}(x) = 1 + \frac{1}{x-\gamma} + O(\frac{1}{x^2})$. This is sufficient to yield simple results like
$$ \sum_{n \le x} \zeta^{-1}(n) = n + \log x -0.9267119... + O\Big(\frac{1}{x}\Big) $$
$$ \sum_{p \le x} \zeta^{-1}(p) = \pi(x) + \log\log x + 0.6332 ... + O\Big(\frac{1}{\log x}\Big) $$
where I have computed the values of their constants after proving their existence. In this example the true value is in finding a complete asymptotic for $\zeta^{-1}(x)$ i.e. what will be the general term of it asymptotic expansion?
So in general, I would like reference to any work done on the inverse of the Riemann Zeta function.
from Hot Weekly Questions - Mathematics Stack Exchange
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