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:

meisselMertens: function [depth][
    Euler: 0.57721566490153286
    m: (1//2) + ln 1-1//2
    loop range.step:2 3 depth 'x ->
        if prime? x ->
            m: m + (1//x) + ln 1-1//x

    return m + Euler
]

print meisselMertens 10000000