I am a college sophomore in US with double majors in mathematics and microbiology. My algorithmic biology research got me passionate about number theory and analysis. I started pursuing a mathematics major this Spring semester. I have been independently self-studying the number theory textbooks written by Niven/Zuckerman/Montgomery, Apostol, and Ireland/Rosen during this semester. As this semester progressed, I discovered that I am more interested in pure mathematics than applied aspects (computational biology, cryptography, etc.). I want to pursue a career as analytic number theorist and prove the Collatz conjecture and Erdos-Straus conjecture.
I have been thinking about doing number-theory research in my university (research university; huge mathematics department). I have been self-studying NT by myself and also regularly attending professional and graduate seminars on number theory. But, I did not do any pure mathematics research as an undergraduate. Should I visit NT professors in my university and ask them if I can do undergraduate research under them? If research is not possible (perhaps due to me lacking maturity), should I request to do independent reading under them and later proceed with research? How should I ask them? What should I address? If even independent reading is not desirable to them, what should I ask them or do by myself?
As for my mathematical background, I have been taking Calculus II (computational) and discrete mathematics. I will be taking Calculus III (vector calc.) over Summer, followed by Analysis I, Probability, Theoretical Linear Algebra in Fall 2015. As for my self-studying on this semester, I have been studying NT textbooks (mentioned above), proof methodologies, and basic linear algebra.