How should LinkedIn summary look like for a person seeking PhD positions? There are many intros, how-to's and guides about writing professional summaries for LinkedIn accounts over the net; however I can not find a good guide to write a summary for academics.

I also tried to write one for myself but I think that this is so vague. Because in the profile we list most of our previous activities such information about our education, career experience, skills, voluntary activities, etc.

If we have detailed information and our CV listed in the profile, what information should explicitly be written in the summary part? Should we summarize the profile in a format like "I have a huge number of publications and thousands of years work experience!"? How brief this part should be? Or is it a bad idea to leave it blank?

What's the purpose of a summary part in a LinkedIn profile for an academic person?

Also, should we mention that we are looking for a PhD position or not? Something like the objectives part in the normal CVs?
Look up some of the professors you are applying to. Unless they work in fields that have strong connections with industry, they will probably not have a profile at all; and if they don't have a profile, they won't even think of looking you up on it.

Even if they do, your application has a lot more information about you, and structured in a way that is more familiar to them (since everybody that applies has an application, but mostly do not have a linkedin account), so it is quicker to look up the information they may interested in.