A Contour–Integral Representation for the *n*-th Prime

We derive a closed‐form analytic representation of the *n*-th prime number $p_n$, valid for all integers $n\ge 2$. The expression consists of a finite elementary backbone followed by a single Mellin–Barnes contour integral whose residues reproduce the classical asymptotic series obtained by inverting the logarithmic integral. Central to the construction is a factorial–free formula for the inversion coefficients $A_k$, expressed in terms of the polylogarithm $\operatorname{Li}_2$ and the gamma function. The divergent remainder of the standard expansion is resummed *via* Borel‐ Mellin techniques, yielding an exact identity rather than an asymptotic equality. Numerical evidence confirms that truncating the contour integral at its first residue already improves the relative error below 0.2 % for $n\gtrsim10^5$.