The inverse Riemann zeta function

TL;DR

This paper introduces a novel formula and method for computing the inverse of the Riemann zeta function across real and complex domains, including detailed zero analysis and a fast computational algorithm.

Contribution

It develops a new analytical approach for the inverse Riemann zeta function, extending to other special functions and providing high-precision numerical methods.

Findings

Formulated an inverse zeta function expression.

Validated convergence of the formulas numerically.

Developed a fast algorithm for inverse zeta computation.

Abstract

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 $w$.