Maybe my question is too vague, but here it goes:
Is it possible to equal a series over prime numbers that has a log in its denominator (for example,
$$\sum_p \frac{1}{p \log p}$$
$$\sum_p \frac{1}{\sqrt{p} \log p}$$
$$\sum_p \frac{1}{ \log p}$$
) to something "useful" as a series over all natural numbers or an expansion involving elementary functions? I know that some of these examples diverge, so my question would be somewhat equivalent to:
Is there any way to expand these series to anything unrelated to prime numbers? If so, how? Is there any general method to treat with that logs?
Thank you.
Edit: After some work with the second series, I have concluded that it is equal to
$$\log \prod_p \frac{1}{1-\frac{1}{\sqrt{p}\log{p}}} + K$$
For a fixed constant $K$. The problem is that those logs prevent me from treating the product as an Euler Product nor translating it to a Dirichlet's Series
You may want to have a look at Abel's Summation Formula.
It is a main tool in Number Theory to obtain asymptotic behaviours and more about these types of series.