Will the prestige of my masters program affect my phd application choices? I am in the last few semesters of my bachelors, and I have been preparing myself for  applying to PhD programs in mathematics. However, as I was sitting with my research mentor today, he informally offered me a graduate assistance-ship for the masters program at my current university. He mentioned I would still need to go through application process but that I am a pure shoe-in for the GA position. I am unsure whether to chase after this offer on a more serious tone or apply for PhD programs, and as such I am seeking information. The real rub stands that I am attending a not too prestigious school for math and this worries me that I may be limiting myself. However, for personal reasons I am considering this potential offer greatly. Would the masters from a low/middle-tier prestige/ranking hinder my chances for getting accepted into top-tier PhD programs? 

If you need some information about me. GPA 3.7, Major GPA 3.8, I will have some published works, I will have finished 3 Graduate courses before applications (Order Theory, Analysis, Graph Theory), GRE general in the upper crust, No Math Subject yet (Still studying for this). Two very strong recommendations from current and accomplished researchers, and two standard recommendations.

There is a note that I should include my location before posting. I am attending MTSU for an Advanced Mathematics Bachelors.
I would recommend that you apply directly to doctoral programs and use this offer as a backup if nothing materializes. In mathematics and some other fields acceptance into doctoral programs with only a BS or BA is more or less the norm. Your vita looks pretty good and I think that many doctoral programs would be happy to have you. 

Most likely you would also be offered some sort of funding, though it would be as a TA or similar, though funded research isn't impossible. Places with doctoral programs in math in the US need a lot of TAs. 

Apply first and then consider your options. You might even be able to get some indication about the likelihood of your acceptance from a faculty member at a proposed university.

Also, in many doctoral programs in the US, you will have the chance to get a Master's along the way. Sometimes just by filling out a request after a certain amount of coursework. In my case a small thesis was also needed.