In this post, we are interested in the Rimenann zeta function $\zeta(s)$ in $s > 1$ only where it is strictly decreasing rather than $s$ in the entire complex plane. We have the Stieltjes series expansion of the Riemann Zeta function. I inverted the first few terms of this series using series reversion and showed that if $s > 1$ and $\zeta(s) = a$, then,
$$ s = \zeta^{-1}(a) = 1 + \frac{1}{a - \gamma_0} - \frac{\gamma_1}{1!(a - \gamma_0)^2} + \frac{\gamma_2}{2!(a - \gamma_0)^3} - \frac{\gamma_3 - 12\gamma_1}{3!(a - \gamma_0)^4} + \mathcal O(a^{-5}) $$
It seems that $\zeta^{-1}(a)$ can be expressed in the form
$$ 1 + \sum_{n=0}^{\infty} (-1)^n\frac{f(\gamma_1, \gamma_2, \ldots, \gamma_n)}{n!(a - \gamma_0)^{n+1}} $$
where $f(\gamma_1, \gamma_2, \ldots, \gamma_n)$ is some polynomial function of the Stieltjes constants $\gamma_n$.
Question: I am looking for a closed or a recurrence formula for $f$? Also any reference in this series in literature?