[notfashion]

Representing an arbitrary sequence of bits, digits, etc. using a real number is the basis for arithmetic coding. As such, it's an approach that has been productive historically, and arithmetic coding continues to be widely used. So the basic idea of encoding whatever you want to encode in a single real number (obviously represented to just enough precision) isn't fruitless and boring. In this case it is a number that is easy to calculate using the formula given, which doesn't at first glance resemble a prime sieve.

[naniwaduni]

It is exactly like arithmetic coding. The formula given allows you to improve the precision of the constant given by encoding more primes, and it equally well lets you encode integers that are not prime. It doesn't resemble a prime sieve because it's not one; you just construct approximations with existing knowledge of what the primes are.

[rlpb]

> In this case it is a number that is easy to calculate using the formula given

It appears to me that the formula given requires pre-existing knowledge of the list of primes in order the expand the series.