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:

What is the renormalized value of the following sums? 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 $c= e$.)