Meissel–Mertens constant

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

Task

Motivation

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.

Example

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 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...

See

Details in the Wikipedia article: Meissel–Mertens constant

PARI/GP

Summation method

{
MM(t)=
  my(s=0);
  forprime(p = 2, t,
    s += log(1.-1./p)+1./p
  );
  Euler+s
};

Output:

Running 10^9 summations to get 9 valid digits:

? \p10
   realprecision = 19 significant digits (10 digits displayed)
? MM(1e9)
?
%1 = 0.2614972129
?
? ##
  ***   last result: cpu time 1min, 18,085 ms, real time 1min, 18,094 ms.
?