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

This Python script calculates a mathematical constant known as "Euler's constant" (sometimes referred to as "Meissel-Mertens constant" or "MM") using a summation formula involving prime numbers. Here's a step-by-step explanation of the code:

1. Importing the math Module:

from math import log

This line imports the log function from the Python math module, which is needed to calculate the natural logarithm.

2. isPrime Function:

def isPrime(n):
   for i in range(2, int(n**0.5) + 1):
       if n % i == 0:
           return False
   return True

This is a function that checks whether a given number n is prime. It does so by iterating through all numbers from 2 to the square root of n (inclusive) to see if n is divisible by any of them. If any such divisor is found, the function returns False, indicating that n is not prime. Otherwise, it returns True.

3. Calculating Euler's Constant:

In the __main__ block, the script initializes a variable Euler with the value of Euler's constant, which is approximately 0.57721566490153286.

It also initializes a variable m to 0, which will be used to accumulate a sum related to prime numbers.

4. Iterating Through Prime Numbers:

The script iterates through numbers x from 2 to 10,000,000 (exclusive) using a for loop. Note that the range is defined as range(2, 10_000_000), where the underscore in 10_000_000 serves as a visual separator for improved readability.

5. Summation of Logarithmic Terms:

For each x that is prime (as determined by the isPrime function), the script calculates a logarithmic term log(1-(1/x)) + (1/x) and adds it to the accumulated sum m. This sum is the main component of the formula used to approximate Euler's constant.

6. Final Calculation and Output:

After the loop completes, the script adds the accumulated value m to the initial value of Euler and assigns the result to a new variable MM. This MM represents the approximated value of Euler's constant.

Finally, the script prints MM to the standard output, displaying the calculated approximation of Euler's constant.

#!/usr/bin/python

from math import log

def isPrime(n):
    for i in range(2, int(n**0.5) + 1):
        if n % i == 0:
            return False        
    return True


if __name__ == '__main__':
    Euler = 0.57721566490153286
    m = 0
    for x in range(2, 10_000_000):
        if isPrime(x):
            m += log(1-(1/x)) + (1/x)

    print("MM =", Euler + m)