Step by Step solution about How to resolve the algorithm Meissel–Mertens constant step by step in the Java programming language

The provided Java code computes the Meissel-Mertens constant, a mathematical constant related to the distribution of prime numbers.

Main Function:

• main method:
  • Generates a list of reciprocals of prime numbers less than 1,000,000,000.
  • Computes the Meissel-Mertens constant by summing the reciprocals of primes and the natural logarithm of (1 - reciprocal) for each prime.
  • Prints the computed constant to the console.

listPrimeReciprocals Method:

• Computes the list of reciprocals of prime numbers up to a specified limit using the Sieve of Eratosthenes algorithm.
• The algorithm maintains a BitSet to track prime numbers efficiently.
• It iterates through prime numbers up to the square root of the limit, marking non-primes in the sieve.
• Finally, it constructs a list of reciprocals for the prime numbers marked in the sieve.

Implementation Details:

• The constant euler represents the Euler-Mascheroni constant.
• The BitSet data structure is used to efficiently track prime numbers.
• Double precision is used to represent the constant and reciprocals to reduce numerical errors.
• The code optimizes the computation by reversing the iteration order for the BitSet.

Mathematical Background: The Meissel-Mertens constant is defined as the sum of the reciprocals of prime numbers. It is named after Ernst Meissel and Franz Mertens, who independently discovered it in the 19th century.

The formula used in the code, euler + sum, is based on the infinite sum representation of the constant:

M = e + lim_{n->∞} Σ(p ≤ n) (1/p + log(1 - 1/p))

where e is the Euler-Mascheroni constant, p is a prime number, and the summation is over all primes less than or equal to n.

import java.util.ArrayList;
import java.util.BitSet;
import java.util.List;

public final class MeisselMertensConstant {
	
	public static void main(String[] aArgs) {
		List<Double> primeReciprocals = listPrimeReciprocals(1_000_000_000);
		final double euler = 0.577_215_664_901_532_861;
		double sum = 0.0;
	    for ( double reciprocal : primeReciprocals ) {
	    	sum += reciprocal + Math.log(1.0 - reciprocal);
	    }	
	    
	    final double meisselMertens = euler + sum;	
	    System.out.println(String.format("%s%.9f", "The Meissel-Mertens constant = ", meisselMertens));
	}	
	
	private static List<Double> listPrimeReciprocals(int aLimit) {
		BitSet sieve = new BitSet(aLimit + 1);
		sieve.set(2, aLimit + 1);
		
		for ( int i = 2; i <= Math.sqrt(aLimit); i = sieve.nextSetBit(i + 1) ) {
			for ( int j = i * i; j <= aLimit; j += i ) {
				sieve.clear(j);
			}
		}
		
		List<Double> result = new ArrayList<Double>(sieve.cardinality());
		for ( int i = 2; i >= 0; i = sieve.nextSetBit(i + 1) ) {
			result.add(1.0 / i);
		}
		
		return result;
	}

}