# application of distribution of primes in arithmetic progressions

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: $$h(x) = f(x) \ln x - \int_{2}^{x} \frac{f(t)}{t} dt$$

• Yes. Because $\ln p =\ln x- \int_p^x \frac{dt}{t}$. As Hamidine said $h(x)$ is a weighted version of $f$, more regular because its Mellin transform is meromorphic. – reuns Apr 4 at 4:02