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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.

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  • $\begingroup$ Asymptotic as what? As $x \to \infty$ (so that $s \to 1^+$)? As $x \to 0^+$ (so that $s \to \infty$)? $\endgroup$ – GEdgar May 27 at 17:44
  • $\begingroup$ If x approches zero then the inverse will nit be injective so that option is eliminated and we are left with only x tends to infinity $\endgroup$ – Nilos May 27 at 18:05
  • $\begingroup$ Right. The other case is $x \to 1^+$ when $s \to \infty$. $\endgroup$ – GEdgar May 28 at 11:43
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    $\begingroup$ Might be related: I had noticed the series reversion of the Stieltjes coefficients expansion gave what seem to be monotonically decreasing inverse coefficients. math.stackexchange.com/questions/2800802/… $\endgroup$ – Benedict W. J. Irwin May 28 at 12:15
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    $\begingroup$ @BenedictW.J.Irwin Very much related ! In fact I had done the series reversion on Stieltjes expansion of $\zeta(s)$ and using it to explore properties of the zeta function. $\endgroup$ – Nilos May 28 at 12:21

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