We know the following:
$$\gamma=\lim_{n\to\infty }\left(\sum_{k=1}^n\frac{1}{k}-\ln(n)\right).$$
This could be a good candidate for renormalized sum of $\left(\sum_{k=1}^{\infty}\frac{1}{k}\right)$.

Also, we know the following:
$$-\gamma=\lim_{n\to\infty }\left(\sum_{p\leq n}\frac{\ln(p)}{p-1}-\ln(n)\right).$$
I want to ask does this analogously mean that $-\gamma$ is renormalized value of $(\sum_{p}\frac{\ln(p)}{p-1})$?


Also, I wanted to ask similar question:

How to assign renormalized value of the following sums? ( Is it even possible?) I.e., is there a way we could assign finite values to the following sums?
\begin{gather}
\tag{1}\label{1}
\sum_{p} \frac{1}{\sqrt{cp}-1} \\
\tag{2}\label{2}
\sum_{p} \frac{\ln(p)}{\sqrt{cp}-1}
\end{gather}
Here $c$ is a constant. (I'm particularly interested in two cases, where $c=1$ and $c= e$.)