is there a faster algorithm to find nth prime?

[winter sundial]

and if so can you explain just the concept or theory of it? I'm having lots of fun implementing the code and want to write it myself.

    static void makeComposite(int[] primeBools, int x) {
        primeBools[x/64] |= (1 << ((x >> 1) & 31));
    }
    public static int isComposite(int[] primeBools, int x) {
        return (primeBools[x/64] & (1 << ((x >> 1) & 31)));
    }
    public static long primeSieve(int n) {
        int estimatedLimit = (int)((Math.log(n) + Math.log(Math.log(n)) - 1 + ( (Math.log(Math.log(n)) - 2) / Math.log(n) )
                - (( Math.pow(Math.log(Math.log(n)),2) - 6 * Math.log(Math.log(n)) - 11 )/( 2 * Math.pow(Math.log(n),2) ))) * n);
        int[] isPrimeBools = new int[estimatedLimit/64 + 1];
        Arrays.fill(isPrimeBools, 0000);
        for (int i = 3; i * i <= estimatedLimit; i+=2) {
            if (isComposite(isPrimeBools, i) == 0) {
                for (int j = i*i, k = i << 1; j < estimatedLimit; j += k) {
                    makeComposite(isPrimeBools, j);
                }
            }
        }
        int finalPrime = 1;
        for (int foundPrimes = 1; foundPrimes <= n; finalPrime += 2) {
            if (isComposite(isPrimeBools, finalPrime) == 0) {
                foundPrimes++;
            }
        }
        return n == 1 ? 2 : finalPrime-2;
    }

[weary grail]

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[winter sundial]

i know i can make it multi-threaded, but that feels more like brute force than a clever solution

[olive geyser]

the prime sieve doesn't find the n-th prime, it finds the highest prime <= n

https://www.baeldung.com/cs/prime-number-algorithms#:~:text=Sieve of Atkin speeds up,more complicated than the others.&text=Comparing this running time with,algorithm for generating prime numbers.

and there are also a few algorithm that give you "probable prime" numbers which are numbers that may be prime and don't have many prime factors (especially not low prime factors).
These probable primes are often used in cryptography