We know the following:

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

This could be the good candidate for regularized 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 regularized value of $(\sum_{p}\frac{\ln(p)}{p-1})$?


Also, I wanted to ask similar question:

What is the regularized value of the following sums in above sense?

1.$$\sum_{p} \frac{1}{\sqrt{cp}-1}$$

2.$$\sum_{p} \frac{\ln(p)}{\sqrt{cp}-1}$$

Here $c$ is a constant. ( I'm particularly interested in two cases where c=1 and c= e)