An Adaptive Zeta-Zero Approximation for the nth Prime with a Certified Error Envelope

An Adaptive Zeta-Zero Approximation for the $n$th Prime with a Certified Error Envelope We investigate an adaptive approximation for the $n$th prime based on the inverse Riemann prime-counting function and oscillatory corrections derived from the nontrivial zeros of the Riemann zeta function. The number of included zeros increases according to $K(n)=\lceil(\log n)^2\rceil$, providing a computationally inexpensive correction without fitted coefficients or enumeration of preceding primes. The proposed estimator is\[\widehat p_n=x_n+2\sqrt{x_n}\sum_{k=1}^{K(n)}\frac{\frac12\cos(\gamma_k\log x_n)+\gamma_k\sin(\gamma_k\log x_n)}{\frac14+\gamma_k^2},\qquad x_n=R^{-1}(n),\]where $\gamma_k$ denotes the imaginary part of the $k$th nontrivial zeta zero on the critical line. Numerical experiments involving the first ten million primes indicate improvements in average prediction accuracy relative to the uncorrected inverse Riemann approximation. An unconditional error envelope is established using published explicit inequalities for the $n$th prime, providing a certified bound throughout a specified range of indices. The construction combines classical explicit-formula methods with an adaptive truncation strategy. The certified bound is inherited from known results and does not establish a new asymptotic error estimate or guarantee improved accuracy for every individual prime. The work is presented as a computational and analytical investigation of adaptive prime approximation.