Summing the Reciprocals of Primes

The sum of the reciprocals of the primes diverges, but very slowly. The sum of the reciprocals of the first 100 primes is 2.106… The sum of the reciprocals of the first 1,000 primes is 2.457… For the first 10,000 it’s 2.709… And it keeps creeping up, ever more slowly. To get the sum to reach 6, you need to add up the reciprocals of the first 3 × 10¹³² primes—far more than the number of atoms in the observable universe! Luckily there is no shortage of primes. Here’s how you can see that the sum diverges, and that it diverges very slowly. First, remember Euler’s product formula for the Riemann zeta function: This diverges logarithmically when Taking logs we see must diverge too—but only log-logarithmically! A deeper result, called Merten’s Second Theorem and proved here, says that for some constant This constant is called the Meissel–Mertens constant. You can think of it as a fancier relative of Euler’s constant which is defined by But it’s much less widespread in mathematics than…