I try to understand an application of distribution of primes in arithmetic progressions
Let $$f(x) = \sum_{p \leq x p\equiv 3 \bmod 10} 1$$
So computing $f(40) = 3$ i.e. the primes: 3, 13, and 23
Now $$h(x) = \sum_{p \leq x p\equiv 3 \bmod 10} \ln p$$
My question is: what is 'the meaning of $h(x)$? and is $h(40)$ equal to: $\ln 3 + \ln 13 + \ln23$
and secondly: why is the relation between $f(x)$ and $h(x)$ equal to: $$h(x) = f(x) \ln x - \int_{2}^{x} \frac{f(t)}{t} dt$$