The Inverse–Li Residue Sieve: A New Local-Analytic, Memory-Light Method for Computing the N-Th Prime

We propose the Inverse–Li Residue Sieve (ILIRS), a novel local-analytic algorithm for com- puting the n-th prime P (n) that: • stores only O(1) floats (no bitmaps, no tables); • needs at most O(log n) deterministic Miller–Rabin tests per index; • relies on minimising RLi(k, n) = Li(k) − n inside a window Θ(√n ln n) around the explicit inverse of Li. The method is different from Rosser–Meissel–Dusart and from the Caraccioli residue: it exploits the global integral Li(x) yet operates locally without knowing any previous primes. Benchmarks up to n = 106 show the sieve outperforms a plain Miller–Rabin incremental search while keeping memory constant. All proofs, code and data are embedded in this file.