convergence divergence - Is there a function that can be subtracted from the sum of reciprocals of primes to make the series convergent

The gamma constant is defined by an equation where the harmonic series is subtracted by the natural logarithm:

$$\gamma = \lim_{n \rightarrow \infty }\left(\sum_{k=1}^n \frac{1}{k} - \ln(n)\right)$$

It is well known that both the harmonic series by itself and the sum of reciprocals of primes are divergent.

Is there a well known function that when subtracted from the sum of reciprocals of primes makes the resultant series convergent?

Is there a function $f(x)$ that makes the following series convergent:

$$\lim_{n \rightarrow \infty }\left(\sum_{p\text{ is a prime }}^n \frac{1}{p} - f(n)\right)$$

Answer

Yes there is a constant associated with the sum of the reciprocals of the primes. In particular, Mertens showed that $$\sum_{p \text{ prime } \le x} \frac1p - \log\log(x)$$ converges to a constant as $x\to \infty$. This is a result from 1874.

I found the result in a paper:

EULER’S CONSTANT: EULER’S WORK AND MODERN DEVELOPMENTS - By JEFFREY C. LAGARIAS

Mertens' paper is titled: Ein Beitrag zur analytischen Zahlentheorie