  In this article, we develop a formula for an inverse Riemann zeta function
such that for $w=\zeta(s)$ we have $s=\zeta^{-1}(w)$ for real and complex
domains $s$ and $w$. The presented work is based on extending the analytical
recurrence formulas for trivial and non-trivial zeros as to solve an equation
$\zeta(s)-w=0$ for a given $w$-domain using logarithmic differentiation and
zeta recursive root extraction methods. We further explore formulas for trivial
and non-trivial zeros of the Riemann zeta function in greater detail, and next,
we also explore an expansion of the inverse zeta function by an attractor of
its branch singularities, and develop some identities that emerge from them. In
the last part, we extend the presented results as a general method for finding
zeros and inverses of many other functions, such as the gamma function, the
Bessel function of the first kind, or finite/infinite degree polynomials and
rational functions, etc. We further compute all the presented formulas
numerically to high precision and show that these formulas do indeed converge
to the inverse of the Riemann zeta function and the related results. We also
develop a fast algorithm to compute $\zeta^{-1}(w)$ for complex $w$.
