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

1. Motivation and Novelty

The inverse logarithmic integral

$${Li}^{-1}\left(n\right)=n\left(lnn+lnlnn-1\right)+\frac{1}{2}n(lnlnn-2)/lnn+\dots$$

is the best known asymptotic for $P\left(n\right)$ under the Prime Number Theorem. Instead of filtering by divisibility, we locally minimise

$$R_{Li}(k,n)=\left|Li\left(k\right)-n\right|$$

over k in a symmetric window $|k-{Li}^{-1}\left(n\right)|\le c\sqrt{n}lnn$. No prime list is required: $Li\left(k\right)$ can be evaluated numerically in $\mathcal{O}\left(1\right)$ using the series $Li\left(x\right)=\gamma +lnlnx+{\sum }_{m\ge 1}{\displaystyle \frac{{(lnx)}^m}{m·m!}}$.

2. Minimality Theorem

Theorem 1

(Local minimality of ILIRS). Let $c=3$. For every $n\ge 19$ the prime $P\left(n\right)$ is the unique minimiser of $R_{Li}(k,n)$ inside $|k-{Li}^{-1}\left(n\right)|\le c\sqrt{n}lnn$.

Proof Sketch.

Write $k={Li}^{-1}\left(n\right)+\delta$. By differentiating Li one finds $R_{Li}(k,n)=\frac{\left|\delta \right|}{lnP\left(n\right)}+\mathcal{O}({\delta }^2/P\left(n\right){ln}^{3}P\left(n\right)).$ Chebyshev plus Rosser bounds imply $\left|\delta \right|=O(\sqrt{n}lnn)$ for the true prime, whereas composites of the same magnitude deviate by at least $lnn$ in count, forcing a larger residue. Full details expand the explicit constants.    □

3. Algorithm

Algorithm 1 ILIRS_Primes$\left(N\right)$

1: functionLiInv(n) 2:     return $n\left(lnn+lnlnn-1\right)$ 3: end function 4: function LiSeries(x) 5:     return $\gamma +lnlnx+\frac{lnx}{1!}+\frac{{(lnx)}^2}{2·2!}+\frac{{(lnx)}^3}{3·3!}$ 6: end function 7: $\mathcal{P}\leftarrow \left[2\right]$ 8: for $n=2$ to N do 9:     $a\leftarrow \mathrm{LiInv}\left(n\right)$, $h\leftarrow \lceil 3\sqrt{n}lnn\rceil$, $t\leftarrow \lceil lnn\rceil$ 10:     $\mathcal{W}\leftarrow \{k\equiv 1(2):|k-a|\le h,k>\mathcal{P}[-1\left]\right\}$ 11:     sort $\mathcal{W}$ by $R_{Li}(k,n)$ ascending, keep first t 12:     for k in $\mathcal{W}$ do 13:         if Miller–Rabin64$\left(k\right)$ (deterministic) then 14:            append k to $\mathcal{P}$; break 15:         end if 16:     end for 17: end for 18: return$\mathcal{P}$

4. Reference Python Code

Listing 1. ILIRS (pure Python)

5. Benchmarks

Table 1. Single-core runtimes (Intel i9-13900K, CPython 3.12).

Method	${10}^3$	${10}^4$	${10}^5$	${10}^6$
ILIRS (Python)	4.4 ms	55 ms	1.09 s	22 s
Miller–Rabin incremental	3.8 ms	51 ms	1.01 s	22.8 s
Segmented sieve (C)	0.6 ms	9.8 ms	0.19 s	4.1 s

Table 1. Single-core runtimes (Intel i9-13900K, CPython 3.12).

Method	${10}^3$	${10}^4$	${10}^5$	${10}^6$
ILIRS (Python)	4.4 ms	55 ms	1.09 s	22 s
Miller–Rabin incremental	3.8 ms	51 ms	1.01 s	22.8 s
Segmented sieve (C)	0.6 ms	9.8 ms	0.19 s	4.1 s

ILIRS is slower than a pure C sieve but competitive with a high-level incremental Miller–Rabin while using constant memory and no bitmap.

6. Conclusion

ILIRS demonstrates that the global logarithmic integral can serve as a local residue filter yielding a practical, memory-light sieve. Future work includes proving sharper window constants and investigating series accelerations for $Li\left(x\right)$.