Problem Statement

Calculate Meissel–Mertens constant up to a precision your language can handle.

Analogous to Euler's constant, which is important in determining the sum of reciprocal natural numbers, Meissel-Mertens' constant is important in calculating the sum of reciprocal primes.

We consider the finite sum of reciprocal natural numbers: 1 + 1/2 + 1/3 + 1/4 + 1/5 ... 1/n this sum can be well approximated with: log(n) + E where E denotes Euler's constant: 0.57721... log(n) denotes the natural logarithm of n.

Now consider the finite sum of reciprocal primes: 1/2 + 1/3 + 1/5 + 1/7 + 1/11 ... 1/p this sum can be well approximated with: log( log(p) ) + M where M denotes Meissel-Mertens constant: 0.26149...

Let's start with the solution:

use v5.36;
use ntheory qw(forprimes);

my $s;
forprimes { $s += log(1 - 1/$_)+1/$_ } 1e9;
say my $result = $s + .57721566490153286;