Few Months Left to Prepare for Pure Math PhD Application
It is hard to do much in a few months and have anything to show for it but effort. But having said that, showing effort may help you a bit. If you have a research opportunity available that will, among other things, let you work closely with a professor or two, it can solve a few of your issues. If you make a strong effort you will not only learn something about research, but have a better chance for good recommendations. 

You don't have to finish the project you start for this to have an effect, but it would be good if it is a project that you are invested in enough that you would want to finish it. It might give you some research direction for the future. 

Beyond that, and even as part of it, ask a lot of questions. Funny thing, but you actually show some intelligence by asking good questions. More so than you would by not asking any and seeming passive around professors. 

I don't think blogs and such will help you much. What you need is effort that will give you some feedback from those who might help you later.